Geometry is one of the sciences, with the application of which, in practice, a person encounters almost every day. Among the variety of geometric shapes, the trapezoid also deserves special attention. It is a convex figure with four sides, two of which are parallel to each other. The latter are called bases, and the remaining two are called sides. The segment perpendicular to the bases and determining the size of the gap between them will be the height of the trapezoid. How can you calculate its length?

Find the height of an arbitrary trapezoid

Based on the initial data, determining the height of the figure is possible in several ways.

Known area

If the length of the parallel sides is known, and the area of ​​the figure is also indicated, then the following relation can be used to determine the desired perpendicular:

S = h * (a + b) / 2,
h - the required value (height),
S is the area of ​​the figure,
a and b are sides parallel to each other.
It follows from the above formula that h = 2S / (a ​​+ b).

The value of the middle line is known

If, among the initial data, in addition to the area of ​​the trapezoid (S), the length of its middle line (l) is also known, then another formula will be useful for calculations. First, it is worth clarifying what the middle line is for this type of quadrangle. The term defines the part of the straight line connecting the midpoints of the sides of the figure.

Based on the trapezoid property l = (a + b) / 2,
l - middle line,
a, b - sides-base of the quadrangle.
Therefore, h = 2S / (a ​​+ b) = S / l.

4 sides of the figure are known

In this case, the Pythagorean theorem will help. Bringing the perpendiculars down to the larger base side, use it for the two resulting right-angled triangles. The final expression will look like:

h = √c 2 - (((a-b) 2 + c 2 -d 2) / 2 (a-b)) 2,

c and d are 2 other sides.

Angles at the base

If you have base angle data, use the trigonometric functions.

h = c * sinα = d * sinβ,

α and β - angles at the base of the quadrangle,
c and d are its sides.

Diagonals of the figure and the angles that they intersect form

Diagonal length - the length of the line segment connecting the opposite vertices of the figure. Let us denote these quantities by the symbols d1 and d2, and the angles between them γ and φ. Then:

h = (d1 * d2) / (a ​​+ b) sin γ = (d1 * d2) / (a ​​+ b) sinφ,

h = (d1 * d2) / 2l sin γ = (d1 * d2) / 2l sinφ,

a and b - sides-base of the figure,
d1 and d2 are the diagonals of the trapezoid,
γ and φ are the angles between the diagonals.

The height of the figure and the radius of the circle that is inscribed in it

As follows from the definition of this kind of circle, it touches each base at 1 point, which are part of one straight line. Therefore, the distance between them - the diameter - is the desired height of the figure. And since the diameter is twice the radius, then:

h = 2 * r,
r is the radius of the circle that was inscribed in this trapezoid.

Find the height of an isosceles trapezoid

• As follows from the formulation, a distinctive characteristic of an isosceles trapezoid is the equality of its lateral sides. Therefore, to find the height of the figure, use the formula to determine this value in the case when the sides of the trapezoid are known.

So, if c = d, then h = √c 2 - (((a-b) 2 + c 2 -d 2) / 2 (a-b)) 2 = √c 2 - (a-b) 2/4,
a, b - sides-base of the quadrangle,
c = d - its sides.

• In the presence of the magnitude of the angles formed by two sides (base and lateral), the height of the trapezoid is determined by the following ratio:

h = c * sinα,
h = c * tgα * cosα = c * tgα * (b - a) / 2c = tgα * (b-a) / 2,

α - angle at the base of the figure,
a, b (a< b) – основания фигуры,
c = d - its sides.

• If the values ​​of the figure's diagonals are given, then the expression for finding the figure's height will change, since d1 = d2:

h = d1 2 / (a ​​+ b) * sinγ = d1 2 / (a ​​+ b) * sinφ,

h = d1 2/2 * l * sinγ = d1 2/2 * l * sinφ.

With such a shape as a trapezoid, we meet in life quite often. For example, any bridge made of concrete blocks is a prime example. A more visual option can be considered the steering of each vehicle and so on. The properties of the figure were known even in Ancient Greece., which Aristotle described in more detail in his scientific work "Beginnings". And the knowledge derived thousands of years ago is still relevant today. Therefore, let's get acquainted with them in more detail.

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Basic concepts

Figure 1. The classic shape of a trapezoid.

A trapezoid is essentially a quadrilateral made up of two line segments that are parallel and two others that are not parallel. When talking about this figure, you should always remember about such concepts as: base, height and middle line. Two segments of a quadrilateral which are called bases to each other (segments AD and BC). Height is called the segment perpendicular to each of the bases (EH), i.e. intersect at an angle of 90 ° (as shown in Figure 1).

If we add up all the internal degree measures, then the sum of the angles of the trapezoid will be equal to 2π (360 °), like any quadrilateral. The segment, the ends of which are the midpoints of the sidewalls (IF) called the middle line. The length of this segment is the sum of the bases BC and AD divided by 2.

There are three types of geometric shapes: straight, regular and isosceles. If at least one angle at the vertices of the base is straight (for example, if ABD = 90 °), then such a quadrangle is called a straight trapezoid. If the side segments are equal (AB and CD), then it is called isosceles (respectively, the angles at the bases are equal).

How to find the area

For, to find the area of ​​a quadrilateral ABCD use the following formula:

Figure 2. Solving the problem of finding an area

For a more illustrative example, let's solve an easy problem. For example, let the upper and lower bases be 16 and 44 cm, respectively, and the sides - 17 and 25 cm. Let's construct a perpendicular segment from the top D in such a way that DE II BC (as shown in Figure 2). Hence we get that

Let DF start up. From ΔADE (which will be isosceles), we get the following:

That is, in simple terms, we first found the height ΔADE, which is also the height of the trapezoid. From here, we calculate the area of ​​the quadrangle ABCD using the already known formula, with the already known value of the height DF.

Hence, the required area ABCD is 450 cm³. That is, we can say with confidence that in order to to calculate the area of ​​the trapezoid, you only need the sum of the bases and the length of the height.

Important! When solving the problem, it is not necessary to find the value of the lengths separately, it is quite acceptable if other parameters of the figure are applied, which, with appropriate proof, will be equal to the sum of the bases.

Types of trapezoids

Depending on which sides the figure has, which angles are formed at the bases, three types of quadrangle are distinguished: rectangular, irregular and isosceles.

Versatile

There are two forms: acute and obtuse... ABCD is acute-angled only when the base angles (AD) are sharp and the side lengths are different. If the value of one angle is Pi / 2 more (the degree measure is more than 90 °), then we get obtuse.

If the sidewalls are equal in length

Figure 3. View of an isosceles trapezoid

If the non-parallel sides are equal in length, then ABCD is called isosceles (regular). Moreover, for such a quadrangle, the degree measure of the angles at the base is the same, their angle will always be less than the right one. It is for this reason that the isosceles are never divided into acute-angled and obtuse-angled. A quadrangle of this shape has its own specific differences, which include:

1. The segments connecting opposite vertices are equal.
2. Sharp angles with a larger base are 45 ° (illustrative example in Figure 3).
3. If you add up the degree measures of opposite angles, then they add up to 180 °.
4. Any regular trapezoid can be built around.
5. If you add up the degree measure of opposite angles, then it is equal to π.

Moreover, due to their geometrical arrangement of points, there exist basic properties of an isosceles trapezoid:

Angle value at base 90 °

The perpendicularity of the side of the base is a capacious characteristic of the concept of "rectangular trapezoid". There cannot be two lateral sides with corners at the base, because otherwise it will already be a rectangle. In this type of quadrangle, the second lateral side will always form an acute angle with a large base, and an obtuse one with a smaller one. In this case, the perpendicular side will also be the height.

The segment between the midpoints of the sidewalls

If you connect the midpoints of the sides, and the resulting segment will be parallel to the bases, and is equal in length to half of their sum, then the formed straight line will be the middle line. The value of this distance is calculated by the formula:

For a more illustrative example, consider the problem with the use of the middle line.

Task. The middle line of the trapezoid is 7 cm, it is known that one of the sides is 4 cm larger than the other (Fig. 4). Find the lengths of the bases.

Figure 4. Solving the problem of finding base lengths

Solution. Let the smaller base DC be equal to x cm, then the larger base will be equal to (x + 4) cm, respectively.From here, using the formula for the middle line of the trapezoid, we get:

It turns out that the smaller DC base is 5 cm, and the larger one is 9 cm.

Important! The concept of the center line is key in solving many problems in geometry. Based on its definition, many proofs for other figures are built. Using the concept in practice, a more rational solution and search for the required value is possible.

Determining the height, and how to find it

As noted earlier, the height is a segment that intersects the bases at an angle of 2Pi / 4 and is the shortest distance between them. Before finding the height of the trapezoid, it is necessary to decide what input values ​​are given. For a better understanding, consider the problem. Find the height of the trapezoid, provided that the bases are 8 and 28 cm, the sides are 12 and 16 cm, respectively.

Figure 5. Solving the problem of finding the height of the trapezoid

Draw segments DF and CH at right angles to the base of AD. By definition, each of them will be the height of a given trapezoid (Fig. 5). In this case, knowing the length of each sidewall, using the Pythagorean theorem, we find what the height in triangles AFD and BHC is equal to.

The sum of the segments AF and HB is equal to the difference between the bases, i.e .:

Let the AF length be equal to x cm, then the length of the segment HB = (20 - x) cm. As found, DF = CH, hence.

Then we get the following equation:

It turns out that the segment AF in the triangle AFD is 7.2 cm, from here we calculate the height of the trapezoid DF by the same Pythagorean theorem:

Those. the height of the trapezoid ADCB will be 9.6 cm. As you can see, the calculation of the height is a more mechanical process, and is based on the calculation of the sides and angles of the triangles. But, in a number of problems in geometry, only the degrees of the angles can be known, in which case the calculations will be made through the aspect ratio of the inner triangles.

Important! In essence, a trapezoid is often thought of as two triangles, or as a combination of a rectangle and a triangle. To solve 90% of all problems encountered in school textbooks, the properties and characteristics of these figures. Most of the formulas for this HMT are derived relying on the "mechanisms" for these two types of figures.

How to quickly calculate base length

Before finding the base of the trapezoid, it is necessary to determine which parameters have already been given, and how to rationally use them. A practical approach is to extract the length of the unknown base from the centerline formula. For a clearer perception of the picture, we will show, using the example of a task, how this can be done. Let it be known that the middle line of the trapezoid is 7 cm, and one of the bases is 10 cm. Find the length of the second base.

Solution: Knowing that the middle line is equal to half the sum of the bases, it can be argued that their sum is 14 cm.

(14 cm = 7 cm × 2). From the condition of the problem, we know that one of them is 10 cm, hence the smaller side of the trapezoid will be 4 cm (4 cm = 14 - 10).

Moreover, for a more comfortable solution to problems of this kind, we recommend that you learn well such formulas from the trapezoid area as:

• middle line;
• square;
• height;
• diagonals.

Knowing the essence (exactly the essence) of these calculations, you can easily find out the desired value.

Video: trapezoid and its properties

Video: trapezoid features

Output

From the examples of tasks considered, one can draw a simple conclusion that the trapezoid, in terms of calculating tasks, is one of the simplest shapes in geometry. To successfully solve problems, first of all, it is not worth deciding what information is known about the object being described, in which formulas they can be applied, and deciding what needs to be found. With this simple algorithm, no problem with this geometric shape is effortless.

A trapezoid is a relief quadrilateral in which two opposite sides are parallel and the other two are not parallel. If all opposite sides of the quadrilateral are pairwise parallel, then this is a parallelogram.

You will need

• - all sides of the trapezoid (AB, BC, CD, DA).

Instructions

1. Non-parallel parties trapeze are called lateral sides, and parallel ones are called bases. The line between the bases, perpendicular to them - height trapeze... If the side parties trapeze are equal, then it is called isosceles. First, consider the solution for trapeze which is not isosceles.

2. Draw line segment BE from point B to the bottom base AD parallel to the side trapeze CD. From the fact that BE and CD are parallel and held between parallel bases trapeze BC and DA, then BCDE is a parallelogram, and its opposite parties BE and CD are equal. BE = CD.

3. Consider the triangle ABE. Calculate the AE side. AE = AD-ED. Foundations trapeze BC and AD are known, and in the parallelogram BCDE the opposite parties 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 semiperimeter. S = root (p * (p-AB) * (p-BE) * (p-AE)). In this formula, p is the semiperimeter of triangle ABE. p = 1/2 * (AB + BE + AE). To calculate the area, you know all the data you need: AB, BE = CD, AE = AD-BC.

6. Express from this formula the height of the triangle, which is also the height trapeze... BH = 2 * S / AE. Calculate it.

7. If the trapezium is isosceles, the solution can be executed differently. Consider triangle ABH. It is rectangular because one of the corners, BHA, is straight.

8. Draw the height CF from the vertex C.

9. Examine the HBCF figure. HBCF rectangle, from the fact that two of it parties- heights, and the other two are bases trapeze, that is, the corners are straight, and the opposite parties are parallel. This means that BC = HF.

10. Look at right-angled triangles ABH and FCD. The angles at heights BHA and CFD are straight, and the angles at the lateral parties x BAH and CDF are equal, because the trapezoid ABCD is isosceles, which means that the triangles are similar. Because the heights BH and CF are equal to either side parties isosceles trapeze AB and CD are equal, then similar triangles are equal. Hence, their parties AH and FD are also equal.

11. Discover AH. AH + FD = AD-HF. Because from the parallelogram HF = BC, and from the triangles AH = FD, then AH = (AD-BC) * 1/2.

A trapezoid is a geometric figure, which is a quadrangle, in which two sides, which are called bases, are parallel, and the other two are not parallel. They are called sides trapeze... The segment drawn through the midpoints of the sides is called the middle line. trapeze... The trapezoid can have different lengths of the sides or identical, 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's much more practical to know how to detect square trapeze .

You will need

• Ruler with millimeter divisions

Instructions

1. Measure all sides trapeze: AB, BC, CD and DA. Write down the results of your measurements.

2. On line AB, sweep the midpoint - point K. On segment DA, sweep out point L, which is also in the middle of segment AD. Combine points K and L, the resulting segment KL will be the middle line trapeze ABCD. Measure line segment KL.

3. From the top trapeze- longing C, lower the perpendicular to its base AD on the segment CE. He will be the height trapeze ABCD. Measure segment CE.

4. We call the segment KL the letter m, and the segment CE the letter h, then square S trapeze Calculate ABCD by the formula: S = m * h, where m is the middle line trapeze ABCD, h - height trapeze ABCD.

5. There is another formula that allows you to calculate square trapeze ABCD. Bottom base trapeze- AD will be called the letter b, and the upper base of BC will be called a. The area is determined by the formula S = 1/2 * (a + b) * h, where a and b are the bases trapeze, h - height trapeze .

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Tip 3: How to find the height of a trapezoid if the area is known

A trapezoid means a quadrilateral in which two of its four sides are parallel to each other. Parallel sides are the foundations of this trapeze, the other two are the lateral sides of this trapeze... Discover the height trapeze, if we know its area, it will be hefty easy.

Instructions

1. You need to figure out how it is allowed to calculate the area of ​​the initial trapeze... 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 trapeze, and h is its height (Height trapeze- perpendicular dropped from one base trapeze to another); S = m * h, where m is the middle line trapeze(The middle line is a segment parallel to the bases trapeze and connecting the middle of its lateral sides).

2. Now, knowing the formulas for calculating the area trapeze, it is allowed to deduce new ones from them, to find the height trapeze: h = (2 * S) / (a ​​+ b); h = S / m.

3. In order to make it clearer how to solve similar problems, it is allowed to see examples: Example 1: Given a trapezoid, whose area is 68 cm ?, the average line of which is 8 cm, you need to find the height given trapeze... In order to solve this problem, you need to use the previously derived formula: h = 68/8 = 8.5 cm Answer: the height of this trapeze is 8.5 cm Example 2: Let the trapeze the area is 120 cm ?, the length of the bases of this trapeze are equal to 8 cm and 12 cm, respectively, it is required to detect the height this trapeze... To do this, you need to apply one of the derived formulas: h = (2 * 120) / (8 + 12) = 240/20 = 12 cm Answer: the height of the given trapeze equal to 12 cm

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Note!
Any trapezoid has a number of properties: - the middle line of a trapezoid is equal to the half-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, it will intersect the point of intersection of the diagonals of the trapezoid; - it is allowed to inscribe a circle in a trapezoid if the sum of the bases of this trapezoid is equal to the sum of its lateral sides. Use these properties when solving problems.

Tip 4: How to find the height of a triangle if the coordinates of the points are given

The height in a triangle is a straight line segment connecting the top of the figure with the opposite side. This segment must certainly be perpendicular to the side, therefore, from each vertex it is allowed to draw only one the height... Due to the fact that there are three peaks in this figure, the heights in it are the same. If a triangle is given by the coordinates of its vertices, it is possible to calculate the length of any of the heights, say, using the formula for finding the area and calculating the lengths of the sides.

Instructions

1. Calculate that the area triangle equal to half the product of the length of each of its sides by the length of the height dropped on 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... Label the coordinates of the vertices of the shape as follows: A (X?, Y?, Z?), B (X?, Y?, Z?) And C (X?, Y?, Z?). Then you can calculate the length of the side AB using the formula AB =? ((X? -X?)? + (Y? -Y?)? + (Z? -Z?)?). For the other 2 parties, 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 to? (15? + 12? + 6?) =? 405? 20.12 and? (2? + 3? + (-6?)) =? 49 = 7.

3. Skills of the lengths of 3 sides obtained in the previous step are sufficient for calculating 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, later substitution of the values ​​obtained from the coordinates triangle example from the previous step, this formula will give this 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. Coming from the 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 by the length of the side to which it is drawn, to find the height, divide the doubled area by the length of the desired side: H = 2 * S / a. For the example used above, the height dropped to the AB side would be 2 * 68.815 / 16.09? 8.55, the height to the BC side will be 2 * 68.815 / 20.12? 6.84, and for the AU side this value will be 2 * 68.815 / 7? 19.66.

A trapezoid is such a quadrangle, two sides of which are parallel (these are the bases of the trapezoid, indicated in the figure a and b), and the other two are not (in the figure, HELL and CB). The height of the trapezoid is the segment h drawn perpendicular to the bases.

How to find the height of the trapezoid with known values ​​of the area of ​​the trapezoid and the lengths of the bases?

To calculate the area S of the trapezoid ABCD, we use the formula:

S = ((a + b) × h) / 2.

Here the segments a and b are the bases of the trapezoid, h is the height of the trapezoid.

Transforming this formula, we can write:

Using this formula, we obtain the value of h if the area S and the lengths of the bases a and b are known.

Example

If it is known that the area of ​​the trapezoid S is 50 cm², the length of the base a is 4 cm, the length of the base b is 6 cm, then to find the height h, we use the formula:

We substitute known values ​​in the formula.

h = (2 × 50) / (4 + 6) = 100/10 = 10 cm

Answer: The height of the trapezoid is 10 cm.

How to find the height of a trapezoid if given the values ​​of the area of ​​the trapezoid and the length of the midline?

Let's use the formula for calculating the area of ​​a trapezoid:

Here m is the middle line, h is the height of the trapezoid.

If the question arises, how to find the height of the trapezoid, the formula:

h = S / m would be the answer.

Thus, we can find the value of the height of the trapezoid h, having the known values ​​of the area S and the segment of the midline m.

Example

We know the length of the middle line of the trapezium m, which is 20 cm, and the area S, which is 200 cm². Let's find the value of the height of the trapezoid h.

Substituting the values ​​of S and m, we get:

h = 200/20 = 10 cm

Answer: The height of the trapezoid is 10 cm

How to find the height of a rectangular trapezoid?

If the trapezoid is a quadrilateral, with two parallel sides (bases) of the trapezoid. That diagonal is a segment that connects two opposite vertices of the corners of the trapezoid (segment AC in the figure). If the trapezoid is rectangular, using the diagonal, we find the height of the trapezoid h.

A rectangular trapezoid is a trapezoid where one of the lateral sides is perpendicular to the bases. In this case, its length (BP) coincides with the height h.

So, consider a rectangular trapezoid ABCD, where AD is the height, DC is the base, AC is the diagonal. Let's use the Pythagorean theorem. The square of the hypotenuse AC of a right-angled triangle ADC is equal to the sum of the squares of its legs AB and BC.

Then you can write:

AC² = AD² + DC².

AD is the leg of the triangle, the side of the trapezium and, at the same time, its height. After all, a segment of blood pressure is perpendicular to the bases. Its length will be:

AD = √ (AC² - DC²)

So, we have a formula for calculating the height of the trapezoid h = AD

Example

If the length of the base of a rectangular trapezoid (DC) is 14 cm, and the diagonal (AC) is 15 cm, we use the Pythagorean theorem to obtain the value of the height (AD-side).

Let x be the unknown leg of a right-angled triangle (AD), then

AC² = AD² + DC² can be written

15² = 14² + x²,

x = √ (15²-14²) = √ (225-196) = √29 cm

Answer: The height of the rectangular trapezoid (AB) will be √29 cm, which is approximately 5.385 cm

How to find the height of an isosceles trapezoid?

An isosceles trapezoid is called a trapezoid, in which the lengths of the sides are equal to each other. A straight line drawn through the midpoints of the bases of such a trapezoid will be the axis of symmetry. A special case is a trapezoid, the diagonals of which are perpendicular to each other, then the height h, will be equal to the half-sum of the bases.

Consider the case if the diagonals are not perpendicular to each other. In an isosceles (isosceles) trapezoid, the angles at the bases are equal and the lengths of the diagonals are equal. It is also known that all vertices of an isosceles trapezoid touch the line of a circle drawn around this trapezoid.

Consider the drawing. ABCD is an isosceles trapezoid. It is known that the bases of the trapezoid are parallel, which means that BC = b parallel to AD = a, side AB = CD = c, which means that the angles at the bases are respectively equal, you can write the angle BAQ = CDS = α, and the angle ABC = BCD = β. Thus, we conclude that triangle ABQ is equal to triangle SCD, which means that the segment

AQ = SD = (AD - BC) / 2 = (a - b) / 2.

Having, according to the condition of the problem, the values ​​of the bases a and b, and the length of the lateral side c, we find the height of the trapezoid h, equal to the segment BQ.

Consider a right-angled triangle ABQ. BO - the height of the trapezoid, perpendicular to the base AD, hence the segment AQ. We find the side AQ of triangle ABQ using the formula we derived earlier:

Having the values ​​of two legs of a right-angled triangle, we find the hypotenuse BQ = h. We use the Pythagorean theorem.

AB² = AQ² + BQ²

Let's substitute these tasks:

c² = AQ² + h².

We obtain the formula for finding the height of an isosceles trapezoid:

h = √ (c²-AQ²).

Example

An isosceles trapezoid ABCD is given, where base AD = a = 10cm, base BC = b = 4cm, and side AB = c = 12cm. Under such conditions, let us consider, for example, how to find the trapezoid height, the isosceles trapezoid AVSD.

Find the side AQ of triangle ABQ by substituting the known data:

AQ = (a - b) / 2 = (10-4) / 2 = 3cm.

Now let's substitute the values ​​of the sides of the triangle into the formula of the Pythagorean theorem.

h = √ (c²- AQ²) = √ (12²- 3²) = √135 = 11.6cm.

Answer. The height h of the isosceles trapezoid ABCD is 11.6 cm.