Trapezoids

Help Questions

Math › Trapezoids

Questions 1 - 10

Which of the following shapes is a trapezoid?

Shapes

Explanation

A trapezoid is a four-sided shape with straight sides that has a pair of opposite parallel sides. The other sides may or may not be parallel. A square and a rectangle are both considered trapezoids.

In the trapezoid below, find the degree measure of .

$76^\circ$

$86^\circ$

$153^\circ$

$93^\circ$

Explanation

In a trapezoid, the angles on the same leg (called adjacent angles) are supplementary, meaning they add up to degrees.

and the angle measuring degrees are adjacent angles that are supplementary.

Given Trapezoid with bases $\overline{TR}$ and $\overline{AP}$ . The trapezoid has midsegment $\overline{MN}$ , where and are the midpoints of $\overline{TP}$ and $\overline{RA}$ , respectively.

True, false, or undetermined: Trapezoid $TRNM \sim$ Trapezoid .

False

True

Undetermined

Explanation

The fact that the trapezoid is isosceles is actually irrelevant. Since and are the midpoints of legs $\overline{TP}$ and $\overline{RA}$ , it holds by definition that

$\overline{TM} \cong \overline{MP}$

and

$\overline{RN} \cong \overline{NA}$

It follows that $\frac{TM}{MP} = \frac{RN}{NA} = 1$

However, the bases of each trapezoid are noncongruent, by definition, so, in particular,

$TR \ne AP$

Assume that $\overline{AP}$ is the longer base - this argument works symmetrically if the opposite is true. Let - equivalently, .

Since the bases are of unequal length, $x \ne 0$ . The length of the midsegment $\overline{MN}$ is the arithmetic mean of the lengths of these two bases, so

$MN = \frac{TR + AP}{2} = \frac{TR + TR + x }{2} = \frac{ 2 \cdot TR + x }{2} = TR + \frac{x}{2}$

Since $TR \ne MN$ , it follows that

$\frac{TR}{MN} \ne 1$ ,

and

$\frac{TR}{MN} \ne \frac{TM}{MP} = \frac{RN}{NA}$ .

This disproves similarity, since one condition is that corresponding sides must be in proportion.

Find the area of the figure.

Explanation

From the figure, you should notice that it is made up of a right triangle and a trapezoid. The lower base of the trapezoid is also the hypotenuse of the right triangle.

First, find the length of the hypotenuse of the right triangle using the Pythagorean Theorem.

$\text{Hypotenuse}=\sqrt{\text{base}^2+\text{height}^2}$

$\text{Hypotenuse}=\sqrt{6^2+12^2}=\sqrt{180}$

Next, use this value to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{1}{2}(\text{base 1}+\text{base 2})(\text{height})$

Plug in the given and found values to find the area.

$\text{Area of Trapezoid}=\frac{1}{2}(10+\sqrt{180})(4)=20+2\sqrt{180}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base})(\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(6)(12)=36$

To find the area of the figure, add the two areas together.

$\text{Area of Figure}=\text{Area of Trapezoid}+\text{Area of Triangle}$

$\text{Area of Figure}=20+2\sqrt{180}+36=82.83$

Make sure to round to places after the decimal.

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:

Which of the following shapes is a trapezoid?

Shapes

Explanation

A trapezoid is a four-sided shape with straight sides that has a pair of opposite parallel sides. The other sides may or may not be parallel. A square and a rectangle are both considered trapezoids.

An isosceles trapezoid has two bases that are parallel to each other. The larger base is times greater than the smaller base. The smaller base has a length of inches and the length of non-parallel sides of the trapezoid have a length of inches.

What is the perimeter of the trapezoid?

Explanation

To find the perimeter of this trapezoid, first find the length of the larger base. Then, find the sum of all of the sides. It's important to note that since this is an isosceles trapezoid, both of the non-parallel sides will have the same length.

The solution is:

The smaller base is equal to inches. Thus, the larger base is equal to:

$2.5\times4=10$

, where the length of one of the non-parallel sides of the isosceles trapezoid.

$P=2.5+10+(2\times 4.5)$

Trapezoid

In the above figure, $\overline{XY}$ is the midsegment of Trapezoid . Give the ratio of the area of Trapezoid to that of Trapezoid .

33 to 19

10 to 3

13 to 6

20 to 13

Explanation

Midsegment $\overline{XY}$ divides Trapezoid into two trapezoids of the same height, which we will call ; the length of the midsegment is half sum of the lengths of the bases:

$XY = \frac{1}{2} (12 + 40) = \frac{1}{2} \cdot 52 = 26$ .

The area of a trapezoid is one half multiplied by its height multiplied by the sum of the lengths of its bases. Therefore, the area of Trapezoid is

$\frac{1}{2} \cdot h \cdot (TR+XY ) = \frac{1}{2} \cdot h \cdot (12+26 ) = \frac{1}{2} \cdot 38 \cdot h = 19 h$

The area of Trapezoid is

$\frac{1}{2} \cdot h \cdot (XY +AP) = \frac{1}{2} \cdot h \cdot ( 26+40 ) = \frac{1}{2} \cdot 66 \cdot h = 33 h$

The ratio of the areas is

$\frac{33 h}{19h} = \frac{33}{19}$ , or 33 to 19.

Trapezoid

In the above figure, $\overline{XY}$ is the midsegment of Trapezoid .

Which is the greater quantity?

(a) Three times the area of Trapezoid

(b) Twice the area of Trapezoid

(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

Midsegment $\overline{XY}$ divides Trapezoid into two trapezoids of the same height, which we will call ; the length of the midsegment is half sum of the lengths of the bases:

$XY = \frac{1}{2} (12 + 36) = \frac{1}{2} \cdot 48= 24$

The area of a trapezoid is one half multiplied by its height multiplied by the sum of the lengths of its bases. Therefore, the area of Trapezoid is

$\frac{1}{2} \cdot h \cdot (TR+XY ) = \frac{1}{2} \cdot h \cdot (12+24) = \frac{1}{2} \cdot 36 \cdot h = 18 h$ .

Three times this is

$3 \cdot 18h = 54 h$ .

The area of Trapezoid is, similarly,

$\frac{1}{2} \cdot h \cdot (XY +AP ) = \frac{1}{2} \cdot h \cdot (24+36 ) = \frac{1}{2} \cdot 60 \cdot h = 30 h$

Twice this is

$2 \cdot 30 h = 60 h$ .

That makes (b) the greater quantity.

Find the area of the figure.

Explanation

From the figure, you should notice that it is made up of a right triangle and a trapezoid. The lower base of the trapezoid is also the hypotenuse of the right triangle.

First, find the length of the hypotenuse of the right triangle using the Pythagorean Theorem.

$\text{Hypotenuse}=\sqrt{\text{base}^2+\text{height}^2}$

$\text{Hypotenuse}=\sqrt{6^2+12^2}=\sqrt{180}$

Next, use this value to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{1}{2}(\text{base 1}+\text{base 2})(\text{height})$

Plug in the given and found values to find the area.

$\text{Area of Trapezoid}=\frac{1}{2}(10+\sqrt{180})(4)=20+2\sqrt{180}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base})(\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(6)(12)=36$

To find the area of the figure, add the two areas together.

$\text{Area of Figure}=\text{Area of Trapezoid}+\text{Area of Triangle}$