Trapezoids - Geometry

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Question

Find the area of a trapezoid with bases of and and a height of .

Answer

The formula for the area of a trapezoid is:

$A = \frac{a + b}{2}h$

Where and are the bases and is the height. Using this formula and the given values, we get:

$A = \frac{8:cm + 12:cm}{2}\left ( 14:cm ) = 133:cm^2$

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Question

What is the area of this regular trapezoid?

Answer

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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Question

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

Answer

Area of a Trapezoid = ½(a+b)h

= ½ (2+6) 4

= ½ (8) * 4

= 4 * 4 = 16

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Question

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?

Answer

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 _a_2. Thus, the area of the square given in the problem is _s_2.

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) = _s_2

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

(4 + s)(s) = 2_s_2

Distribute the s on the left.

4_s_ + _s_2 = 2_s_2

Subtract _s_2 from both sides.

4_s_ = _s_2

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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Question

What is the area of the following trapezoid?

Answer

The formula for the area of a trapezoid is:

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

where $b_{1}$ is the value of the top base, $b_{2}$ is value of the bottom base, and is the value of the height.

Plugging in our values, we get:

$A = \frac{1}{2} (8: m + 10: m)(4: m)$

$A = \frac{1}{2} (18: m)(4: m)$

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Question

Which of the following shapes is a trapezoid?

Answer

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.

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Question

What is the area of the trapezoid pictured above in square units?

Answer

The formula for the area of a trapezoid is the average of the bases times the height,

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

Looking at this problem and when the appropriate values are plugged in, the formula yields:

$A=(\frac{10+24}{2})7$

$A=\frac{34}{2}\cdot 7$

$A=\frac{238}{2}=119$

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Question

What is the height of the trapezoid pictured above?

Answer

To find the height, we must introduce two variables, $b_{1}, b_{2}$ , each representing the bases of the triangles on the outside, so that $b_{1}+b_{2}+11=25, b_{1}+b_{2}=14$ . (Equation 1)

The next step is to set up two Pythagorean Theorems,

$13^2=b_{1}^2+h^2, 15^2=b_{2}^2+h^2$ (Equation 2, 3)

The next step is a substitution from the first equation,

$b_{1}=14-b_{2}, b_{1}^2=196-28b_{2}+b_{2}^2$ (Equation 4)

and plugging it in to the second equation, yielding

$13^2=196-28b_{2}+b_{2}^2+h^2$ (Equation 5)

The next step is to substitute from Equation 3 into equation 5,

$h^2=15^2-b_{2}^2$ , $169=196-28b_{2}+b_{2}^2+225-b_{2}^2$ which simplifies to

$-252=-28b_{2}, b_{2}=9$

Once we have one of the bases, just plug into the Pythagorean Theorem,

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Question

A isosceles trapezoid with sides , , , and has a height of , what is the area?

Answer

An isosceles trapezoid has two sides that are the same length and those are not the bases, so the bases are 10 and 20.

The area of the trapezoid then is:

$A=\frac{b_{1}+b_{2}}{2}h=\frac{10+20}{2}12=15 \cdot 12=180$

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Question

If the height of a trapezoid is , bottom base is , and the top base is , what is the area?

Answer

The formula for finding the area of a trapezoid is:

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

Substitute the given values to find the area.

$A=\frac{1}{2}(20+10)(10)$

$A= \frac{1}{2}(30)(10)$

$A=\frac{1}{2}(300)$

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Question

Find the area of a trapezoid with bases of length and and a height of .

Answer

The formula for the area of a trapezoid is:

$A = \frac{a + b}{2}h$

Where and are the bases and is the height. Using this formula and the given values, we get:

$A = \frac{12:cm + 15:cm}{2}\left ( 7:cm) = 94.5:cm^2$

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Question

Find the area of the above trapezoid.

Answer

The formula for the area of a trapezoid is:

$A = \frac{a + b}{2}h$

Where and are the bases and is the height. Using this formula and the given values, we get:

$A = \frac{13:units + 29:units}{2}\left ( 15:units ) = 315:units^2$

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Question

Find the area of the above trapezoid.

Answer

The formula for the area of a trapezoid is:

$A = \frac{a + b}{2}h$

Where and are the bases and is the height. Using this formula and the given values, we get:

$A = \frac{4:units + 10:units}{2}\left ( 4:units\right ) = 28:units^2$

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Question

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

Answer

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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Question

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

is

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

Answer

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

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

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

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Question

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.

Answer

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

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

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

$49+h^{2}=625$

$h^{2}=576$

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

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

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

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

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Question

Show algebraically how to develop the trapezoid area formula.

Answer

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

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

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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Question

Find the area of the trapezoid.

Answer

Using the trapezoid area formula, $A = \frac{h}{2}(b_{1} + b_{2})$ ,

height , base $b_{1} = a$ , base $b_{2} = 4a$ .

After substition, the resulting expression is $= \frac{2a}{2}(a + 4a)$ .

The resulting expression is then simplified,

$= 5a^{2}$

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Question

Find the area of the shaded region.

Answer

The shaded region is between the outer and inner trapezoid. To find the area of the shaded region, subtract the area of the inner trapezoid from the area of the outer trapezoid.

Area of the shaded region = $A_{S}$ , area of the outer trapezoid = $A_{O}$ , area of the inner trapezoid = $A_{I}$ .

$A_{O} = \frac{5}{2}(5 + 10)$ , $A_{I} = \frac{3}{2}(4 + 6)$

$A_{S} = A_{O} - A_{I}$

$= \frac{5}{2}(5 + 10) - \frac{3}{2}(4 + 6)$

$= \frac{5}{2}(15) - \frac{3}{2}(10)$

$= \frac{75}{2} - \frac{30}{2}$

$= \frac{45}{2}$

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Question

What is the area of this trapezoid?

Answer

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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