How to find the height of a trapezoid if the bases are known. How to find the height of a trapezoid
The practice of last year's Unified State Exam and State Examination shows that geometry problems cause difficulties for many schoolchildren. You can easily cope with them if you memorize all the necessary formulas and practice solving problems.
In this article you will see formulas for finding the area of a trapezoid, as well as examples of problems with solutions. You may come across the same ones in KIMs during certification exams or at Olympiads. Therefore, treat them carefully.
What you need to know about the trapezoid?
To begin with, let us remember that trapezoid is called a quadrilateral in which two opposite sides, also called bases, are parallel, and the other two are not.
In a trapezoid, the height (perpendicular to the base) can also be lowered. The middle line is drawn - this is a straight line that is parallel to the bases and equal to half of their sum. As well as diagonals that can intersect, forming acute and obtuse angles. Or, in some cases, at a right angle. In addition, if the trapezoid is isosceles, a circle can be inscribed in it. And describe a circle around it.
Trapezoid area formulas
First, let's look at the standard formulas for finding the area of a trapezoid. We will consider ways to calculate the area of isosceles and curvilinear trapezoids below.
So, imagine that you have a trapezoid with bases a and b, in which height h is lowered to the larger base. Calculating the area of a figure in this case is as easy as shelling pears. You just need to divide the sum of the lengths of the bases by two and multiply the result by the height: S = 1/2(a + b)*h.
Let's take another case: suppose in a trapezoid, in addition to the height, there is a middle line m. We know the formula for finding the length of the middle line: m = 1/2(a + b). Therefore, we can rightfully simplify the formula for the area of a trapezoid to the following form: S = m*h. In other words, to find the area of a trapezoid, you need to multiply the center line by the height.
Let's consider another option: the trapezoid contains diagonals d 1 and d 2, which do not intersect at right angles α. To calculate the area of such a trapezoid, you need to divide the product of the diagonals by two and multiply the result by the sin of the angle between them: S= 1/2d 1 d 2 *sinα.
Now consider the formula for finding the area of a trapezoid if nothing is known about it except the lengths of all its sides: a, b, c and d. This is a cumbersome and complex formula, but it will be useful for you to remember it just in case: S = 1/2(a + b) * √c 2 – ((1/2(b – a)) * ((b – a) 2 + c 2 – d 2)) 2.
By the way, the above examples are also true for the case when you need the formula for the area of a rectangular trapezoid. This is a trapezoid, the side of which adjoins the bases at a right angle.
Isosceles trapezoid
A trapezoid whose sides are equal is called isosceles. We will consider several options for the formula for the area of an isosceles trapezoid.
First option: for the case when a circle with radius r is inscribed inside an isosceles trapezoid, and the side and larger base form sharp cornerα. A circle can be inscribed in a trapezoid provided that the sum of the lengths of its bases is equal to the sum of the lengths of the sides.
The area of an isosceles trapezoid is calculated as follows: multiply the square of the radius of the inscribed circle by four and divide it all by sinα: S = 4r 2 /sinα. Another area formula is a special case for the option when the angle between the large base and the side is 30 0: S = 8r2.
Second option: this time we take an isosceles trapezoid, in which in addition the diagonals d 1 and d 2 are drawn, as well as the height h. If the diagonals of a trapezoid are mutually perpendicular, the height is half the sum of the bases: h = 1/2(a + b). Knowing this, it is easy to transform the formula for the area of a trapezoid already familiar to you into this form: S = h 2.
Formula for the area of a curved trapezoid
Let's start by figuring out what a curved trapezoid is. Imagine a coordinate axis and a graph of a continuous and non-negative function f that does not change sign within a given segment on the x-axis. A curvilinear trapezoid is formed by the graph of the function y = f(x) - at the top, the x axis is at the bottom (segment), and on the sides - straight lines drawn between points a and b and the graph of the function.
It is impossible to calculate the area of such a non-standard figure using the above methods. Here you need to apply mathematical analysis and use the integral. Namely: the Newton-Leibniz formula - S = ∫ b a f(x)dx = F(x)│ b a = F(b) – F(a). In this formula, F is the antiderivative of our function on the selected segment. And the area of a curvilinear trapezoid corresponds to the increment of the antiderivative on a given segment.
Sample problems
To make all these formulas easier to understand in your head, here are some examples of problems for finding the area of a trapezoid. It will be best if you first try to solve the problems yourself, and only then compare the answer you receive with the ready-made solution.
Task #1: Given a trapezoid. Its larger base is 11 cm, the smaller one is 4 cm. The trapezoid has diagonals, one 12 cm long, the second 9 cm.
Solution: Construct a trapezoid AMRS. Draw a straight line РХ through vertex P so that it is parallel to the diagonal MC and intersects the straight line AC at point X. You will get a triangle APХ.
We will consider two figures obtained as a result of these manipulations: triangle APX and parallelogram CMRX.
Thanks to the parallelogram, we learn that PX = MC = 12 cm and CX = MR = 4 cm. From where we can calculate the side AX of the triangle ARX: AX = AC + CX = 11 + 4 = 15 cm.
We can also prove that the triangle APX is right-angled (to do this, apply the Pythagorean theorem - AX 2 = AP 2 + PX 2). And calculate its area: S APX = 1/2(AP * PX) = 1/2(9 * 12) = 54 cm 2.
Next you will need to prove that triangles AMP and PCX are equal in area. The basis will be the equality of the parties MR and CX (already proven above). And also the heights that you lower on these sides - they are equal to the height of the AMRS trapezoid.
All this will allow you to say that S AMPC = S APX = 54 cm 2.
Task #2: The trapezoid KRMS is given. On its lateral sides there are points O and E, while OE and KS are parallel. It is also known that the areas of trapezoids ORME and OKSE are in the ratio 1:5. RM = a and KS = b. You need to find OE.
Solution: Draw a line parallel to RK through point M, and designate the point of its intersection with OE as T. A is the point of intersection of a line drawn through point E parallel to RK with the base KS.
Let's introduce one more notation - OE = x. And also the height h 1 for the triangle TME and the height h 2 for the triangle AEC (you can independently prove the similarity of these triangles).
We will assume that b > a. The areas of the trapezoids ORME and OKSE are in the ratio 1:5, which gives us the right to create the following equation: (x + a) * h 1 = 1/5(b + x) * h 2. Let's transform and get: h 1 / h 2 = 1/5 * ((b + x)/(x + a)).
Since the triangles TME and AEC are similar, we have h 1 / h 2 = (x – a)/(b – x). Let’s combine both entries and get: (x – a)/(b – x) = 1/5 * ((b + x)/(x + a)) ↔ 5(x – a)(x + a) = (b + x)(b – x) ↔ 5(x 2 – a 2) = (b 2 – x 2) ↔ 6x 2 = b 2 + 5a 2 ↔ x = √(5a 2 + b 2)/6.
Thus, OE = x = √(5a 2 + b 2)/6.
Conclusion
Geometry is not the easiest of sciences, but you can certainly cope with the exam questions. It is enough to show a little perseverance in preparation. And, of course, remember all the necessary formulas.
We tried to collect all the formulas for calculating the area of a trapezoid in one place so that you can use them when you prepare for exams and revise the material.
Be sure to tell your classmates and friends about this article. in social networks. Let good grades there will be more for the Unified State Examination and State Examination Test!
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Area of a trapezoid. Greetings! In this publication we will look at this formula. Why is she exactly like this and how to understand her. If there is understanding, then you don’t need to teach it. If you just want to look at this formula and urgently, then you can immediately scroll down the page))
Now in detail and in order.
A trapezoid is a quadrilateral, two sides of this quadrilateral are parallel, the other two are not. Those that are not parallel are the bases of the trapezoid. The other two are called sides.
If the sides are equal, then the trapezoid is called isosceles. If one of the sides is perpendicular to the bases, then such a trapezoid is called rectangular.
In its classic form, a trapezoid is depicted as follows - the larger base is at the bottom, respectively, the smaller one is at the top. But no one forbids depicting her and vice versa. Here are the sketches:
Next important concept.
The midline of a trapezoid is a segment that connects the midpoints of the sides. The middle line is parallel to the bases of the trapezoid and equal to their half-sum.
Now let's delve deeper. Why is this so?
Consider a trapezoid with bases a and b and with the middle line l, and perform some additional constructions: draw straight lines through the bases, and perpendiculars through the ends of the midline until they intersect with the bases:
*Letter designations for vertices and other points are not included intentionally to avoid unnecessary designations.
Look, triangles 1 and 2 are equal according to the second sign of equality of triangles, triangles 3 and 4 are the same. From the equality of triangles follows the equality of the elements, namely the legs (they are indicated in blue and red, respectively).
Now attention! If we mentally “cut off” the blue and red segments from the lower base, then we will be left with a segment (this is the side of the rectangle) equal to the middle line. Next, if we “glue” the cut blue and red segments to the upper base of the trapezoid, then we will also get a segment (this is also the side of the rectangle) equal to the midline of the trapezoid.
Got it? It turns out that the sum of the bases will be equal to the two middle lines of the trapezoid:
View another explanation
Let's do the following - construct a straight line passing through the lower base of the trapezoid and a straight line that will pass through points A and B:
We get triangles 1 and 2, they are equal along the side and adjacent angles (the second sign of equality of triangles). This means that the resulting segment (in the sketch it is indicated in blue) is equal to the upper base of the trapezoid.
Now consider the triangle:
*The midline of this trapezoid and the midline of the triangle coincide.
It is known that a triangle is equal to half of the base parallel to it, that is:
Okay, we figured it out. Now about the area of the trapezoid.
Trapezoid area formula:
They say: the area of a trapezoid is equal to the product of half the sum of its bases and height.
That is, it turns out that it is equal to the product of the center line and the height:
You've probably already noticed that this is obvious. Geometrically, this can be expressed this way: if we mentally cut off triangles 2 and 4 from the trapezoid and place them on triangles 1 and 3, respectively:
Then we will get a rectangle with an area equal to the area of our trapezoid. The area of this rectangle will be equal to the product of the center line and the height, that is, we can write:
But the point here is not in writing, of course, but in understanding.
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That's all. Good luck to you!
Sincerely, Alexander.
A trapezoid is a relief quadrilateral in which two opposite sides are parallel and the other two are non-parallel. If all opposite sides of a quadrilateral are parallel in pairs, then it is a parallelogram.
You will need
- – all sides of the trapezoid (AB, BC, CD, DA).
Instructions
1. Non-parallel sides trapezoids are called lateral sides, and parallel sides are called bases. The line between the bases, perpendicular to them - height trapezoids. If lateral sides trapezoids are equal, then it is called isosceles. First, let's look at the solution for trapezoids, which is not isosceles.
2. Draw line segment BE from point B to the lower base AD parallel to the side trapezoids CD. Because BE and CD are parallel and drawn between parallel bases trapezoids BC and DA, then BCDE is a parallelogram, and its opposite sides BE and CD are equal. BE=CD.
3. Look at the triangle ABE. Calculate side AE. AE=AD-ED. Grounds trapezoids BC and AD are known, and in a parallelogram BCDE are opposite sides ED and BC are equal. ED=BC, so AE=AD-BC.
4. Now find out the area of triangle ABE using Heron's formula by calculating the semi-perimeter. S=root(p*(p-AB)*(p-BE)*(p-AE)). In this formula, p is the semi-perimeter of triangle ABE. p=1/2*(AB+BE+AE). To calculate the area, you know all the necessary data: AB, BE=CD, AE=AD-BC.
6. Express from this formula the height of the triangle, which is also the height trapezoids. BH=2*S/AE. Calculate it.
7. If the trapezoid is isosceles, the solution can be executed differently. Look at the triangle ABH. It is rectangular because one of the corners, BHA, is right.
8. Draw height CF from vertex C.
9. Study the HBCF figure. HBCF rectangle, because there are two of it sides are heights, and the other two are bases trapezoids, that is, the angles are right, and the opposite sides parallel. This means that BC=HF.
10. Look at the right triangles ABH and FCD. The angles at heights BHA and CFD are right, and the angles at lateral sides x BAH and CDF are equal because the trapezoid ABCD is isosceles, which means the triangles are similar. Because the heights BH and CF are equal or lateral sides isosceles trapezoids AB and CD are congruent, then similar triangles are congruent. So they sides AH and FD are also equal.
11. Discover AH. AH+FD=AD-HF. Because from a parallelogram HF=BC, and from triangles AH=FD, then AH=(AD-BC)*1/2.
Trapezoid – geometric figure, which is a quadrilateral in which two sides, called bases, are parallel, and the other two are not parallel. They are called sides trapezoids. The segment drawn through the midpoints of the lateral sides is called the midline trapezoids. A trapezoid can have different side lengths or identical ones, in which case it is called isosceles. If one of the sides is perpendicular to the base, then the trapezoid will be rectangular. But it is much more practical to know how to detect square trapezoids .
You will need
- Ruler with millimeter divisions
Instructions
1. Measure all sides trapezoids: AB, BC, CD and DA. Record your measurements.
2. On segment AB, mark the middle - point K. On segment DA, mark point L, which is also located in the middle of segment AD. Combine points K and L, the resulting segment KL will be the middle line trapezoids ABCD. Measure the segment KL.
3. From the top trapezoids– toss C, lower the perpendicular to its base AD on the segment CE. It will be the height trapezoids ABCD. Measure the segment CE.
4. Let us call the segment KL the letter m, and the segment CE the letter h, then square S trapezoids ABCD is calculated using the formula: S=m*h, where m is the middle line trapezoids ABCD, h – height trapezoids ABCD.
5. There is another formula that allows you to calculate square trapezoids ABCD. Bottom base trapezoids– Let’s call AD the letter b, and the upper base BC the letter a. The area is determined by the formula S=1/2*(a+b)*h, where a and b are the bases trapezoids, h – height trapezoids .
Video on the topic
Tip 3: How to find the height of a trapezoid if the area is known
A trapezoid is a quadrilateral in which two of its four sides are parallel to each other. Parallel sides are the bases of this trapezoids, the other two are the lateral sides of this trapezoids. Discover height trapezoids, if you know its area, it will be very easy.
Instructions
1. We need to figure out how to calculate the area of the initial trapezoids. There are several formulas for this, depending on the initial data: S = ((a+b)*h)/2, where a and b are the lengths of the bases trapezoids, and h is its height (Height trapezoids– perpendicular, lowered from one base trapezoids to another);S = m*h, where m is the middle line trapezoids(The middle line is a segment parallel to the bases trapezoids and connecting the midpoints of its sides).
2. Now, knowing the formulas for calculating area trapezoids, it is allowed to derive new ones from them to find the height trapezoids:h = (2*S)/(a+b);h = S/m.
3. In order to make it clearer how to solve similar problems, you can look at examples: Example 1: Given a trapezoid whose area is 68 cm?, the middle line of which is 8 cm, you need to find height given trapezoids. In order to decide this task, you need to use the previously derived formula: h = 68/8 = 8.5 cm Answer: the height of this trapezoids is 8.5 cmExample 2: Let y trapezoids area is 120 cm?, the length of the bases is given trapezoids are equal to 8 cm and 12 cm respectively, it is required to detect height this trapezoids. To do this, you need to apply one of the derived formulas:h = (2*120)/(8+12) = 240/20 = 12 cmAnswer: height of the given trapezoids equal to 12 cm
Video on the topic
Note!
Any trapezoid has a number of properties: - the middle line of a trapezoid is equal to half the sum of its bases; - the segment that connects the diagonals of the trapezoid is equal to half the difference of its bases; - if a straight line is drawn through the midpoints of the bases, then it will intersect the point of intersection of the diagonals of the trapezoid; - You can inscribe a circle into a trapezoid if the sum of the bases of a given trapezoid is equal to the sum of its sides. Use these properties when solving problems.
Tip 4: How to find the height of a triangle given the coordinates of the points
The height in a triangle is the straight line segment connecting the vertex of the figure to the opposite side. This segment must necessarily be perpendicular to the side; therefore, from any vertex it is allowed to draw only one height. Because there are three vertices in this figure, there are the same number of heights. If a triangle is given by the coordinates of its vertices, the length of each of the heights can be calculated, say, using the formula for finding the area and calculating the lengths of the sides.
Instructions
1. Proceed in your calculations from the fact that the area triangle is equal to half the product of the length of each of its sides by the length of the height lowered onto this side. From this definition it follows that to find the height you need to know the area of the figure and the length of the side.
2. Start by calculating the lengths of the sides triangle. Designate the coordinates of the vertices of the figure as follows: A(X?,Y?,Z?), B(X?,Y?,Z?) and C(X?,Y?,Z?). Then you can calculate the length of side AB using the formula AB = ?((X?-X?)? + (Y?-Y?)? + (Z?-Z?)?). For the other 2 sides, these formulas will look like this: BC = ?((X?-X?)? + (Y?-Y?)? + (Z?-Z?)?) and AC = ?((X ?-X?)? + (Y?-Y?)? + (Z?-Z?)?). Let's say for triangle with coordinates A(3,5,7), B(16,14,19) and C(1,2,13) the length of side AB will be?((3-16)? + (5-14)? + (7 -19)?) = ?(-13? + (-9?) + (-12?)) = ?(169 + 81 + 144) = ?394 ? 19.85. The lengths of the sides BC and AC, calculated by the same method, will be equal?(15? + 12? + 6?) = ?405? 20.12 and?(2? + 3? + (-6?)) =?49 = 7.
3. Knowing the lengths of 3 sides obtained in the previous step is enough to calculate the area triangle(S) according to Heron’s formula: S = ? * ?((AB+BC+CA) * (BC+CA-AB) * (AB+CA-BC) * (AB+BC-CA)). Let's say, after substituting into this formula the values obtained from the coordinates triangle-example from the previous step, this formula will give the following value: S = ?*?((19.85+20.12+7) * (20.12+7-19.85) * (19.85+7-20 .12) * (19.85+20.12-7)) = ?*?(46.97 * 7.27 * 6.73 * 32.97) ? ?*?75768.55 ? ?*275.26 = 68.815.
4. Based on area triangle, calculated in the previous step, and the lengths of the sides obtained in the second step, calculate the heights for each of the sides. Because the area is equal to half the product of the height and the length of the side to which it is drawn, to find the height, divide the doubled area by the length of the required side: H = 2*S/a. For the example used above, the height lowered to side AB will be 2*68.815/16.09? 8.55, the height to the BC side will have a length of 2*68.815/20.12? 6.84, and for the AC side this value will be equal to 2*68.815/7? 19.66.
A trapezoid is a convex quadrilateral in which two opposite sides are parallel and the other two are non-parallel. If all opposite sides of a quadrilateral are parallel in pairs, then it is a parallelogram.
You will need
- - all sides of the trapezoid (AB, BC, CD, DA).
Instructions
- Non-parallel sides trapezoids are called laterals, and parallel ones are called bases. The line between the bases, perpendicular to them - height trapezoids. If the sides trapezoids are equal, then it is called isosceles. First let's look at the solution for trapezoids, which is not isosceles.
- Draw line segment BE from point B to the lower base AD parallel to the side trapezoids CD. Since BE and CD are parallel and drawn between parallel bases trapezoids BC and DA, then BCDE is a parallelogram and its opposite sides BE and CD are equal. BE=CD.
- Consider triangle ABE. Calculate side AE. AE=AD-ED. Grounds trapezoids BC and AD are known, and in parallelogram BCDE the opposite sides ED and BC are equal. ED=BC, so AE=AD-BC.
- Now find out the area of triangle ABE using Heron's formula by calculating the semi-perimeter. S=root(p*(p-AB)*(p-BE)*(p-AE)). In this formula, p is the semi-perimeter of triangle ABE. p=1/2*(AB+BE+AE). To calculate the area, you know all the necessary data: AB, BE=CD, AE=AD-BC.
- Next, write down the area of triangle ABE in a different way - it is equal to half the product of the height of triangle BH and the side AE to which it is drawn. S=1/2*BH*AE.
- Express from this formula height triangle, which is also the height trapezoids. BH=2*S/AE. Calculate it.
- Swipe from vertex C height CF.
- Study the HBCF figure. HBCF is a rectangle because two of its sides are heights and the other two are bases trapezoids, that is, the angles are right and the opposite sides are parallel. This means that BC=HF.
- Look at the right triangles ABH and FCD. The angles at the heights BHA and CFD are right, and the angles at the sides BAH and CDF are equal, since the trapezoid ABCD is isosceles, which means the triangles are similar. Since the heights BH and CF are equal or the lateral sides of an isosceles trapezoids AB and CD are congruent, then similar triangles are congruent. This means that their sides AH and FD are also equal.
- Find AH. AH+FD=AD-HF. Since from a parallelogram HF=BC, and from triangles AH=FD, then AH=(AD-BC)*1/2.
- Next, from the right triangle ABH, using the Pythagorean theorem, calculate height B.H. The square of the hypotenuse AB is equal to the sum of the squares of the legs AH and BH. BH=root(AB*AB-AH*AH).
If the trapezoid is isosceles, the solution can be done differently. Consider triangle ABH. It is rectangular because one of the corners, BHA, is right.
The many-sided trapezoid... It can be arbitrary, isosceles or rectangular. And in each case you need to know how to find the area of a trapezoid. Of course, the easiest way is to remember the basic formulas. But sometimes it’s easier to use one that is derived taking into account all the features of a particular geometric figure.
A few words about the trapezoid and its elements
Any quadrilateral whose two sides are parallel can be called a trapezoid. In general, they are not equal and are called bases. The larger one is the lower one, and the other one is the upper one.
The other two sides turn out to be lateral. In an arbitrary trapezoid they have different lengths. If they are equal, then the figure becomes isosceles.
If suddenly the angle between any side and the base turns out to be equal to 90 degrees, then the trapezoid is rectangular.
All these features can help in solving the problem of how to find the area of a trapezoid.
Among the elements of the figure that may be indispensable in solving problems, we can highlight the following:
- height, that is, a segment perpendicular to both bases;
- the middle line, which has at its ends the midpoints of the lateral sides.
What formula can be used to calculate the area if the base and height are known?
This expression is given as a basic one because most often one can recognize these quantities even when they are not given explicitly. So, to understand how to find the area of a trapezoid, you will need to add both bases and divide them by two. Then multiply the resulting value by the height value.
If we designate the bases as a 1 and a 2, and the height as n, then the formula for the area will look like this:
S = ((a 1 + a 2)/2)*n.
The formula that calculates the area if its height and center line are given
If you look carefully at the previous formula, it is easy to notice that it clearly contains the value of the midline. Namely, the sum of the bases divided by two. Let the middle line be designated by the letter l, then the formula for the area becomes:
S = l * n.
Ability to find area using diagonals
This method will help if the angle formed by them is known. Suppose that the diagonals are designated by the letters d 1 and d 2, and the angles between them are α and β. Then the formula for how to find the area of a trapezoid will be written as follows:
S = ((d 1 * d 2)/2) * sin α.
You can easily replace α with β in this expression. The result will not change.
How to find out the area if all sides of the figure are known?
There are also situations when exactly the sides of this figure are known. This formula is cumbersome and difficult to remember. But probably. Let the sides have the designation: a 1 and a 2, the base a 1 is greater than a 2. Then the area formula will take the following form:
S = ((a 1 + a 2) / 2) * √ (in 1 2 - [(a 1 - a 2) 2 + in 1 2 - in 2 2) / (2 * (a 1 - a 2)) ] 2 ).
Methods for calculating the area of an isosceles trapezoid
The first is due to the fact that a circle can be inscribed in it. And, knowing its radius (it is denoted by the letter r), as well as the angle at the base - γ, you can use the following formula:
S = (4 * r 2) / sin γ.
The last general formula, which is based on knowledge of all sides of the figure, will be significantly simplified due to the fact that the sides have the same meaning:
S = ((a 1 + a 2) / 2) * √ (in 2 - [(a 1 - a 2) 2 / (2 * (a 1 - a 2))] 2 ).
Methods for calculating the area of a rectangular trapezoid
It is clear that any of the above is suitable for any figure. But sometimes it is useful to know about one feature of such a trapezoid. It lies in the fact that the difference between the squares of the lengths of the diagonals is equal to the difference made up of the squares of the bases.
Often the formulas for a trapezoid are forgotten, while the expressions for the areas of a rectangle and triangle are remembered. Then you can use a simple method. Divide the trapezoid into two shapes, if it is rectangular, or three. One will definitely be a rectangle, and the second, or the remaining two, will be triangles. After calculating the areas of these figures, all that remains is to add them up.
This is a fairly simple way to find the area of a rectangular trapezoid.
What if the coordinates of the vertices of the trapezoid are known?
In this case, you will need to use an expression that allows you to determine the distance between points. It can be applied three times: in order to find out both bases and one height. And then just apply the first formula, which is described a little higher.
To illustrate this method, the following example can be given. Given vertices with coordinates A(5; 7), B(8; 7), C(10; 1), D(1; 1). You need to find out the area of the figure.
Before finding the area of the trapezoid, you need to calculate the lengths of the bases from the coordinates. You will need the following formula:
length of the segment = √((difference of the first coordinates of the points) 2 + (difference of the second coordinates of the points) 2 ).
The upper base is designated AB, which means its length will be equal to √((8-5) 2 + (7-7) 2 ) = √9 = 3. The lower one is CD = √ ((10-1) 2 + (1-1 ) 2 ) = √81 = 9.
Now you need to draw the height from the top to the base. Let its beginning be at point A. The end of the segment will be on the lower base at the point with coordinates (5; 1), let this be point H. The length of the segment AN will be equal to √((5-5) 2 + (7-1) 2 ) = √36 = 6.
All that remains is to substitute the resulting values into the formula for the area of a trapezoid:
S = ((3 + 9) / 2) * 6 = 36.
The problem was solved without units of measurement, because the scale of the coordinate grid was not specified. It can be either a millimeter or a meter.
Sample problems
No. 1. Condition. The angle between the diagonals of an arbitrary trapezoid is known; it is equal to 30 degrees. The smaller diagonal has a value of 3 dm, and the second is 2 times larger. It is necessary to calculate the area of the trapezoid.
Solution. First you need to find out the length of the second diagonal, because without this it will not be possible to calculate the answer. It is not difficult to calculate, 3 * 2 = 6 (dm).
Now you need to use the appropriate formula for area:
S = ((3 * 6) / 2) * sin 30º = 18/2 * ½ = 4.5 (dm 2). The problem is solved.
Answer: The area of the trapezoid is 4.5 dm2.
No. 2. Condition. In the trapezoid ABCD, the bases are the segments AD and BC. Point E is the middle of the SD side. A perpendicular to straight line AB is drawn from it, the end of this segment is designated by the letter H. It is known that the lengths AB and EH are equal to 5 and 4 cm, respectively. It is necessary to calculate the area of the trapezoid.
Solution. First you need to make a drawing. Since the value of the perpendicular is less than the side to which it is drawn, the trapezoid will be slightly elongated upward. So EH will be inside the figure.
To clearly see the progress of solving the problem, you will need to perform additional construction. Namely, draw a straight line that will be parallel to side AB. The points of intersection of this line with AD are P, and with the continuation of BC are X. The resulting figure VHRA is a parallelogram. Moreover, its area is equal to the required one. This is due to the fact that the triangles that were obtained during additional construction are equal. This follows from the equality of the side and two angles adjacent to it, one vertical, the other lying crosswise.
You can find the area of a parallelogram using a formula that contains the product of the side and the height lowered onto it.
Thus, the area of the trapezoid is 5 * 4 = 20 cm 2.
Answer: S = 20 cm 2.
No. 3. Condition. The elements of an isosceles trapezoid have the following values: lower base - 14 cm, upper - 4 cm, acute angle - 45º. You need to calculate its area.
Solution. Let the smaller base be designated BC. The height drawn from point B will be called VH. Since the angle is 45º, triangle ABH will be rectangular and isosceles. So AN=VN. Moreover, AN is very easy to find. It is equal to half the difference in bases. That is (14 - 4) / 2 = 10 / 2 = 5 (cm).
The bases are known, the heights are calculated. You can use the first formula, which was discussed here for an arbitrary trapezoid.
S = ((14 + 4) / 2) * 5 = 18/2 * 5 = 9 * 5 = 45 (cm 2).
Answer: The required area is 45 cm 2.
No. 4. Condition. There is an arbitrary trapezoid ABCD. Points O and E are taken on its lateral sides, so that OE is parallel to the base of AD. The area of the AOED trapezoid is five times larger than that of the OVSE. Calculate the OE value if the lengths of the bases are known.
Solution. You will need to draw two parallel lines AB: the first through point C, its intersection with OE - point T; the second through E and the point of intersection with AD will be M.
Let the unknown OE=x. The height of the smaller trapezoid OVSE is n 1, the larger AOED is n 2.
Since the areas of these two trapezoids are related as 1 to 5, we can write the following equality:
(x + a 2) * n 1 = 1/5 (x + a 1) * n 2
n 1 / n 2 = (x + a 1) / (5 (x + a 2)).
The heights and sides of the triangles are proportional by construction. Therefore, we can write one more equality:
n 1 / n 2 = (x - a 2) / (a 1 - x).
In the last two entries on the left side there are equal values, which means that we can write that (x + a 1) / (5(x + a 2)) is equal to (x - a 2) / (a 1 - x).
A number of transformations are required here. First multiply crosswise. Parentheses will appear to indicate the difference of squares, after applying this formula you will get a short equation.
In it you need to open the brackets and move all the terms with the unknown “x” to the left, and then extract the square root.
Answer: x = √ ((a 1 2 + 5 a 2 2) / 6).
