How to find the length of the side of a trapezoid

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Math › How to find the length of the side of a trapezoid

Questions 1 - 10

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 .

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

Trapezoid 1

The above figure depicts Trapezoid with midsegment $\overline{MN }$ . Express in terms of .

Explanation

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

The correct choice is .

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 .

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

Trapezoid 1

Figure NOT drawn to scale.

The above figure depicts Trapezoid with midsegment $\overline{MN}$ . , and .

Give the area of Trapezoid in terms of .

Explanation

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

Therefore,

The area of Trapezoid is one half multiplied by the height, , multiplied by the sum of the lengths of the bases, and . The midsegment of a trapezoid bisects both legs, so , and the area is

$A = \frac{1}{2} \cdot MP \cdot (MN + AP)$

$A = \frac{1}{2} \cdot 6 \cdot (20 + 40 - x )$

$A =3 \cdot (60 - x )$

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.

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

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 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 .

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

Trapezoid

In the above diagram, which depicts Trapezoid , and . Which is the greater quantity?

(a)

(b) 24

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

Explanation

To see that (b) is the greater quantity of the two, it suffices to construct the midsegment of the trapezoid - the segment which has as its endpoints the midpoints of legs $\overline{TP }$ and $\overline{RA }$ . Since and , the midsegment, $\overline{MN }$ , is positioned as follows: