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Advanced Geometry : Trapezoids

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Example Question #1 : Trapezoids

A trapezoid has a base of length 4, another base of length s, and a height of length s. A square has sides of length s. What is the value of s such that the area of the trapezoid and the area of the square are equal?

Possible Answers:

$\sqrt{2}$

$2\sqrt{2}$

Correct answer:

Explanation:

In general, the formula for the area of a trapezoid is (1/2)(a + b)(h), where a and b are the lengths of the bases, and h is the length of the height. Thus, we can write the area for the trapezoid given in the problem as follows:

area of trapezoid = (1/2)(4 + s)(s)

Similarly, the area of a square with sides of length a is given by a². Thus, the area of the square given in the problem is s².

We now can set the area of the trapezoid equal to the area of the square and solve for s.

(1/2)(4 + s)(s) = s²

Multiply both sides by 2 to eliminate the 1/2.

(4 + s)(s) = 2s²

Distribute the s on the left.

4s + s² = 2s²

Subtract s² from both sides.

4s = s²

Because s must be a positive number, we can divide both sides by s.

4 = s

This means the value of s must be 4.

The answer is 4.

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Example Question #1 : How To Find The Area Of A Trapezoid

What is the area of this regular trapezoid?

Screen_shot_2013-03-18_at_3.27.27_pm

Possible Answers:

Correct answer:

Explanation:

To solve this question, you must divide the trapezoid into a rectangle and two right triangles. Using the Pythagorean Theorem, you would calculate the height of the triangle which is 4. The dimensions of the rectangle are 5 and 4, hence the area will be 20. The base of the triangle is 3 and the height of the triangle is 4. The area of one triangle is 6. Hence the total area will be 20+6+6=32. If you forget to split the shape into a rectangle and TWO triangles, or if you add the dimensions of the trapezoid, you could arrive at 26 as your answer.

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Example Question #1 : How To Find The Area Of A Trapezoid

Trap

What is the area of the trapezoid above if a = 2, b = 6, and h = 4?

Possible Answers:

Correct answer:

Explanation:

Area of a Trapezoid = ½(a+b)*h

= ½ (2+6) * 4

= ½ (8) * 4

= 4 * 4 = 16

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Example Question #1 : How To Find The Area Of A Trapezoid

Find the area of a trapezoid if the height is , and the small and large bases are and , respectively.

Possible Answers:

16x

12x^2

6x^2

12x

Correct answer:

6x^2

Explanation:

Write the formula to find the area of a trapezoid.

$A= \frac{1}{2} (b1+b2)h$

Substitute the givens and evaluate the area.

$A= \frac{1}{2} (b1+b2)h=\frac{1}{2} (2x+4x)(2x) = 2x^2+4x^2=6x^2$

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Example Question #53 : Quadrilaterals

A right triangle and rectangle are placed adjacent to one another such that the composite figure formed by the triangle and rectangle is a trapezoid.

Find the area of the trapezoid given that the base of the triangle is 7 ft and the hypotenuse of the triangle is 25ft. The base of the rectangle is 9 feet.

Possible Answers:

$312.5 ft^{2}$

$144ft^{2}$

$300ft^{2}$

$2880 ft^{2}$

Correct answer:

$300ft^{2}$

Explanation:

$First\ we\ can\ find\ the\ height\ of\ the\ right\ triangle\ and\ assume\ that\ this\ is\ also\ the\ height\ of\ the\ rectangle\ and\ the\ trapezoid.$

$The\ height\ of\ the\ right\ triangle\ is\ obtained\ by\ using\ the\ Pythagorean\ theorem.$

$7^{2}+h^{2}=25^{2}$

$49+h^{2}=625$

$h^{2}=576$

h=24ft

$We\ know\ the\ top\ of\ the\ trapezoid\ is\ the\ same\ as\ the\ base\ of\ the\ rectangle\ which\ is\ 9\ cm.$

$We\ know\ that\ the\ base\ of\ the\ trapezoid\ is\ the\ sum\ of\ the\ base\ of\ the\ triangle\ and\ the\ base\ of\ the\ rectangle: 9+7=16$

$Now\ we\ have\ all\ parts\ needed\ to\ plug\ into\ the\ formula\ for\ the\ area\ of\ a\ trapezoid: A=\frac{b_{1}+b^{_{2}}}{2}*h$

$A=\frac{9+16}{2}*24=300cm^{2}$

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Example Question #351 : Advanced Geometry

The average of the lengths of the top and bottom of trapezoid

ABCD is 15 cm.

What is the height of the trapezoid, if the area is $180 cm^{2}$ ?

Possible Answers:

9 cm

12 cm

18 cm

6 cm

Correct answer:

12 cm

Explanation:

$The\ area\ formula\ for\ a\ trapezoid\ is\ \frac{(b_{1}+b_{2})}2{}*h.$

$Adding\ the\ base\ and\ height\ and\ dividing\ by\ 2\ is\ the\ same\ as\ taking\ the\ average\ of\ the\ top\ and\ bottom.$ $So\ we\ can\ substitute\ 15\ cm\ for\ the\ \frac{(b_{1}+b_{2})}2{}\ portion\ of\ the\ formula.$ $We\ can\ set\ 15h=180\ and\ solve\ for\ h\ to\ get\ the\ height.$

$Divide\ both\ sides\ by\ 15\ to\ find\ that\ h=12cm.$

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Example Question #51 : Plane Geometry

What is the area of this trapezoid?

Screen shot 2015 10 21 at 3.29.16 pm

Possible Answers:

29.25

44.5

Correct answer:

29.25

Explanation:

The area of a trapezoid is

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

In this particular case the known values are as follows.

$\\b_1=5 \\b_2=8 \\h=4.5$

Substituting in the given values into the area formula we arrive at the following solution.

$\frac{1}{2}(5+8)4.5=29.25$

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Example Question #11 : How To Find The Area Of A Trapezoid

$\small \small Find\ the\ height\ of\ a\ trapezoid\ with\ bases\ of\ 8\ and\ 12\ and\ an\ area\ of\ 80m^{2}.$

Possible Answers:

$\small \small \small 10\ m$

$\small 60\ m$

$\small \small 8\ m$

$\small 12\ m$

Correct answer:

$\small \small 8\ m$

Explanation:

$\small Use\ the\ area\ formula\ for\ a\ trapezoid.$

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

$\small Substitute\ each\ known\ value\ then\ use\ algebra\ to\ solve\ for\ the\ height.$

$\small \small 80=\frac{8+12}{2}h$

$\small 80=10h$

$\small 8=h$

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Example Question #61 : Plane Geometry

Find the area of the trapezoid given below:

Trap1

Possible Answers:

$300\ cm^2$

$325\ cm^2$

$240\ cm^2$

$260\ cm^2$

Correct answer:

$300\ cm^2$

Explanation:

In order to find the area of a trapezoid, we must use the formula below:

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

$In\ this\ trapezoid\ b_{1}=30\ and\ b_{2}=20.$

$We\ can\ also\ see\ that\ h=12.$

$When\ we\ plug\ in\ all\ of\ the\ above\ values\ into\ the\ formula\ we\ get:$

$A=\frac{20+30}{2}(12)$

A=300cm^2

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Example Question #11 : How To Find The Area Of A Trapezoid

Show algebraically how to develop the trapezoid area formula.

Varsity7

Possible Answers:

$h(b_{1}\cdot b_{2})$

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

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

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

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

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

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

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

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

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

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

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

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

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

Correct answer:

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

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

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

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

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

Explanation:

1) The area $A_{1}$ of the triangle that has base $b_{1}$ is $A_{1} = \frac{1}{2}(b_{1})h$ .

2) The area $A_{2}$ of the triangle that has base $b_{2}$ is $A_{2} = \frac{1}{2}(b_{2})h$ .

3) The sum of the two triangel areas is

$A_{1} + A_{2}$

$= \frac{1}{2}b_{1}h + \frac{1}{2}b_{2}h$

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

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

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

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

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Advanced Geometry : Trapezoids

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #1 : Trapezoids

Example Question #1 : How To Find The Area Of A Trapezoid

Example Question #1 : How To Find The Area Of A Trapezoid

Example Question #1 : How To Find The Area Of A Trapezoid

Example Question #53 : Quadrilaterals

Example Question #351 : Advanced Geometry

Example Question #51 : Plane Geometry

Example Question #11 : How To Find The Area Of A Trapezoid

Example Question #61 : Plane Geometry

Example Question #11 : How To Find The Area Of A Trapezoid

All Advanced Geometry Resources

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