Geometry - How to find the length of the side of a trapezoid

An isosceles trapezoid has base measurements of and . The perimeter of the trapezoid is . Find the length for one of the two remaining sides.

Explanation

To solve this problem, first note that an isosceles trapezoid has two parallel bases that are nonequivalent in length. Additionally, an isosceles trapezoid must have two nonparallel sides that have equivalent lengths. Since this problem provides the length for both of the bases as well as the total perimeter, the missing sides can be found using the following formula: Perimeter= Base one Base two (leg), where the length of "leg" is one of the two equivalent nonparallel sides.

Thus, the solution is:

$leg=\frac{10}{2}=5$

Check the solution by plugging in the answer:

Find the value of if the area of this trapezoid is .

Explanation

The formula to find the area of a trapezoid is

$\text{Area}=\frac{1}{2}(base_1+base_2)(height)$ .

Substitute in the values for the area, a base, and the height. Then solve for .

$90=\frac{1}{2}(14+a)6$

An isosceles trapezoid has base measurements of and . Additionally, the isosceles trapezoid has a height of . Find the length for one of the two missing sides.

$\sqrt{20}$

$\sqrt{16}$

$2\sqrt{2}$

$2\sqrt{20}$

Explanation

In order to solve this problem, first note that an isosceles trapezoid has two parallel bases that are nonequivalent in length. Additionally, an isosceles trapezoid must have two nonparallel sides that have equivalent lengths.

This problem provides the lengths for each of the bases as well as the height of the isosceles trapezoid. In order to find the length for one of the two equivalent nonparallel legs of the trapezoid, first use the height of the trapezoid to form right triangles on the interior of the trapezoid that each have a base length of . See image below:

Isos. trap intermediate geo
Note: the base length of can be found by subtracting the lengths of the two bases, then dividing that difference in half:

$4\div2=2$

Now, apply the formula , where the length for one of the two equivalent nonparallel legs of the trapezoid.

Thus, the solution is:

$c=\sqrt{20}$

Find the value of if the area of this trapezoid is .

Explanation

The formula to find the area of a trapezoid is

$\text{Area}=\frac{1}{2}(base_1+base_2)(height)$ .

Substitute in the values for the area, a base, and the height. Then solve for .

$130=\frac{1}{2}(17+b)5$

Suppose the area of the trapezoid is , with a height of and a base of . What must be the length of the other base?

Explanation

Write the formula for finding the area of a trapezoid.

$A=\frac{h}{2}(b_1+b_2)$

Substitute the givens and solve for either base.

$30=\frac{10}{2}(5+b_2)$

Find the value of if the area of the trapezoid below is .

Explanation

The formula to find the area of a trapezoid is

$\text{Area}=\frac{1}{2}(base_1+base_2)(height)$ .

Substitute in the values for the area, a base, and the height. Then solve for .

$60=\frac{1}{2}(12+c)6$

Isos. trap intermediate geo

The isosceles trapezoid shown above has base measurements of and . Additionally, the trapezoid has a height of . Find the length of side .

$3\sqrt{26}$

$\sqrt{26}$

$9\sqrt{26}$

$\sqrt{15}$

$3\sqrt{15}$

Explanation

In this problem the lengths for each of the bases and the height of the isosceles trapezoid is provided in the question prompt. In order to find the length for one of the two equivalent nonparallel legs of the trapezoid (side ), first use the height of the trapezoid to form right triangles on the interior of the trapezoid that each have a base length of .

The base of the interior triangles is equal to because the difference between the two bases is equal to . And, this difference must be divided evenly in half because the isosceles trapezoid is symmetric--due to the two equivalent nonparallel sides and the two nonequivalent parallel bases.

Now, apply the pythagorean theorem: , where .

$c=\sqrt{234}=\sqrt{9\times 26}=\sqrt{9}\sqrt{26}=3\sqrt{26}$

Thus, $x=3\sqrt{26}$

Find the area of the following trapezoid.

Geo_area2

$128 ft^{2}$

$152 ft^{2}$

$190 ft^{2}$

$96 ft^{2}$

$160 ft^{2}$

Explanation

The correct answer is 128 sq ft.

There are two ways to find the total area. One way to find the total area, you must find the area of the triangle and rectangle separately. After some deduction , you can find that the base of the triangle is 6 ft. Then using the Pythagorean Theorem, or 3-4-5 right triangles, you can find that the height of the triangle and rectangle is 8 ft. Geo_area2b

To find the area of the triangle, you would multiply 6 by 8 and then divide by 2 to get 24. To find the area of the rectangle, you would multiply 8 by 13 to get 104. Then you would add both areas to get 128 sq ft.

Geo_area2c

The other way to find the area is to use the formula for area of a trapezoid. After some deduction , you can find that the base of the triangle is 6ft. Then using the Pythagorean Theorem or 3-4-5 right triangles, you can find that the height of the triangle and rectangle is 8 ft.

Then you use the formula:

${\frac{1}{2}}h (b_{1} +b_{2})$

to get

$\frac{1}{2}\cdot 8(13+19)=128sq ft$

Find the value of if the area of the trapezoid below is .

Explanation

The formula to find the area of a trapezoid is

$\text{Area}=\frac{1}{2}(base_1+base_2)(height)$ .

Substitute in the values for the area, a base, and the height. Then solve for .

$210=\frac{1}{2}(18+d)14$

Vt custom trap.

Using the trapezoid shown above, find the length of side .

$2\sqrt{29}$

$\sqrt{29}$

Explanation

In order to find the length of side , first note that the vertical side that has a length of and the base side with length must be perpendicular because they form a right angle. This means that the height of the trapezoid must equal . A right triangle can be formed on the interior of the trapezoid that has a height of and a base lenght of The base length can be derived by finding the difference between the two nonequivalent parallel bases.