You will need

- calculator

Instruction

1

Trapezoid is a quadrilateral with two sides parallel to the ground, and the other two not parallel sides. Trapezoid, the sides of which are equal, is called isosceles or ravnovesnoi. If in an isosceles trapezoid the diagonals are perpendicular, then the height is equal to the sum of bases, we consider the case when diagonals are not perpendicular.

2

Consider an isosceles trapezoid ABCD and describe its properties, but only those, knowledge of which will help us to solve the problem. From the definition of an isosceles trapezoid, the base AD = a is parallel to BC = b, and side AB = CD = c, it follows that the angles at the bases equal, that is, the angle BAQ = CDS = α, in the same way, the angle ABC = BCD = β. Summing up, it is fair to say that the triangle ABQ is equal to the triangle SCD, and hence, cut off AQ = SD = (AD – BC)/2 = (a – b)/2.

3

If in the problem statement we are given the lengths of the bases a and b and length of side C, the height h of the trapezoid is equal to the segment BQ is as follows. Consider the triangle ABQ, because, by definition, the height of the trapezoid is perpendicular to the base, it can be argued that the triangle ABQ rectangular. Side of the AQ triangle ABQ, based on the properties of an isosceles trapezoid, is given by AQ = (a – b)/2. Now knowing the two sides AQ and c by the Pythagorean theorem, find the height h. The Pythagorean theorem States that the hypotenuse squared is equal to the sum of the squares of the legs. We write this theorem applied to our problem: c^2=AQ^2+ h^2. It follows that h = √(c^2-AQ^2).

4

For example, consider the trapezoid ABCD in which the base AD = a = 10cm BC = b = 4 cm, side AB = c = 12cm. To find the height of a trapezoid h. Find the side of the AQ triangle ABQ. AQ = (a – b)/2 = (10-4)/2=3cm. Then substitute the values of the sides of the triangle in the Pythagorean theorem. h = √(c^2-AQ^2) = √(12^2-3^2) =√135=11.6 see

Useful advice

Properties of an isosceles trapezoid.

A straight line passing through the middle of the bases perpendicular to the bases is the axis of symmetry of the trapezoid.

Height, lowered from the top on the larger base, divides it into two segments, one of which is equal to the sum of the bases, the other half-difference basis.

In an isosceles trapezoid the angles at either base are equal.

In an isosceles trapezoid the length of the diagonals.

About an isosceles trapezoid can be circumscribed by a circle.

If in an isosceles trapezoid the diagonals are perpendicular, then the height is equal to the sum of the bases.

A straight line passing through the middle of the bases perpendicular to the bases is the axis of symmetry of the trapezoid.

Height, lowered from the top on the larger base, divides it into two segments, one of which is equal to the sum of the bases, the other half-difference basis.

In an isosceles trapezoid the angles at either base are equal.

In an isosceles trapezoid the length of the diagonals.

About an isosceles trapezoid can be circumscribed by a circle.

If in an isosceles trapezoid the diagonals are perpendicular, then the height is equal to the sum of the bases.

# Advice 2: How to find the area of an isosceles trapezoid

An isosceles trapezoid is a trapezoid whose nonparallel opposite sides are equal. Some formulas allow to find the area of a trapezoid with its sides, angles, height, etc. For the case of isosceles trapezoids the formula may be somewhat simplified.

You will need

- The formula for the area of a regular trapezoid

Instruction

1

The most common formula for calculating area of a trapezoid is S = (a+b)h/2. For the case of an isosceles trapezoid it explicitly does not change. We can only note that an isosceles trapezoid the angles at any of the bases are equal (DAB = CDA = x). As its sides is equal (AB = CD = C), then the height h can be calculate by the formula h = C*sin(x).

Then S = (a+b)*C*sin(x)/2.

Similarly, the area of a trapezoid can be recorded through the secondary side of the trapezoid: S = mh.

Then S = (a+b)*C*sin(x)/2.

Similarly, the area of a trapezoid can be recorded through the secondary side of the trapezoid: S = mh.

2

Consider a special case of an isosceles trapezoid, when its diagonals are perpendicular. In this case, on the property of a trapezoid, the height is equal to the sum of the bases.

Then the area of a trapezoid can be calculated by the formula: S = (a+b)^2/4.

Then the area of a trapezoid can be calculated by the formula: S = (a+b)^2/4.

3

Consider also another formula to find the area of the trapezoid: S = ((a+b)/2)*sqrt(c^2 - ((b-a)^2+c^2-d^2)/2(b-a))^2), where c and d are the sides of a trapezoid. Then, in the case of an isosceles trapezoid, when c = d, the formula becomes: S = ((a+b)/2)*sqrt(c^2-((b-a)^2/2(b-a))^2).

# Advice 3: How to find the diagonal of an isosceles trapezoid

Trapezoid, where the length of sides equal and base parallel, is called isosceles or isosceles. Both diagonals in such a geometrical figure have the same length, which is depending on known parameters of the trapezoid can be calculated many different ways.

Instruction

1

If you know the lengths of the bases of an isosceles trapezoid (A and B) and the length of its sides (C), to determine the lengths of the diagonals (D) we can use the fact that the sum of the squares of the lengths of all sides equals the sum of the squares of the lengths of the diagonals. This property stems from the fact that each of the diagonals of a trapezoid is the hypotenuse of a triangle, the legs of which serve as sidewall and the base. And according to Pythagoras the sum of the squares of the lengths of the legs is equal to the squared length of the hypotenuse. As the sides in an isosceles trapezoid are equal and its diagonals, then this property can be written in the form: A2 + B2 + 2C2 = 2D2. From this formula it follows that the length of the diagonal is equal to the square root of half the sum of the squares of the lengths of the bases, folded with the square of the length of the sides: D = √((A2 + B2)/2 + C2).

2

If the lengths of the sides is not known, but is the length of the middle line (L) and height (H) of an isosceles trapezoid, the length of the diagonal (D) is also easy to calculate. Since the length of the middle line is equal to the sum of the bases of the trapezoid, it gives us the opportunity to find the length of a line between a point on a larger basis, which lowered the height, and the top adjacent to the ground. In an isosceles trapezoid the length of this segment will coincide with the length of the middle line. Since the diagonal closes this segment and the height of the trapezoid into a right triangle, calculate its length is not difficult. For example, according to the same Pythagorean theorem it is equal to the square root of the sum of the squares of the height and the middle line: D=√(L2 + H2).

3

If you know the lengths of both bases of an isosceles trapezoid (A and B) and its height (H), then, as in the previous case, we can calculate the length of a line between point lowered on the big side elevation and the adjacent top. The formula from the previous step transformirovalsya like the following: D=√((A + B)2/4 + H2).