Plane Geometry - Math

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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

This figure is an isosceles trapezoid with bases of 6 in and 18 in and a side of 10 in.

What is the area of the isoceles trapezoid?

Answer

In order to find the area of an isoceles trapezoid, you must average the bases and multiply by the height.

The average of the bases is straight forward:

$\frac{6+18}{2}=12$

In order to find the height, you must draw an altitude. This creates a right triangle in which one of the legs is also the height of the trapezoid. You may recognize the Pythagorean triple (6-8-10) and easily identify the height as 8. Otherwise, use $a^{2}+b^{2}=c^{2}$ .

$36+b^{2}=100$

$b^{2}=64$

Multiply the average of the bases (12) by the height (8) to get an area of 96.

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Question

Find 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 is the length of one base, is the length of the other base, and is the height.

To find the height of the trapezoid, use a Pythagorean triple:

Plugging in our values, we get:

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

$A = \frac{1}{2}(16m+26m)(12m)=252m^2$

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Question

Find the area of the following trapezoid:

Answer

Use the formula for triangles in order to find the length of the bottom base and the height.

The formula is:

$a-a\sqrt{3}-2a$

Where is the length of the side opposite the $\measuredangle 30$ .

Beginning with the side, if we were to create a triangle, the length of the base is $6\sqrt{3}m$ , and the height is .

Creating another triangle on the left, we find the height is , the length of the base is $2\sqrt{3}m$ , and the side is $4\sqrt{3}m$ .

The formula for the area of a trapezoid is:

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

Where is the length of one base, is the length of the other base, and is the height.

Plugging in our values, we get:

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

$A = \frac{1}{2}(16m+16m+6\sqrt{3}m+2\sqrt{3}m)(6m)$

$A = \frac{1}{2}(32m+8\sqrt{3}m)(6m)$

$A = (32m+8\sqrt{3}m)(3m)=96+24\sqrt{3}m^2$

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Question

Determine 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 is the length of one base, is the length of another base, and is the length of the height.

Plugging in our values, we get:

$A=\frac{1}{2}(7m+5m)(4m)=24m^2$

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Question

Find 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 is the length of one base, is the length of another base, and is the length of the height.

Use the Pythagorean Theorem to find the height of the trapezoid:

$B=2\sqrt{91}m$

Plugging in our values, we get:

$A=\frac{1}{2}(18m+30m)(2\sqrt{91}m)=48\sqrt{91}m^2$

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Question

Find the area of the following trapezoid:

Answer

The formula for the area of a trapezoid is

.

Use the Pythagorean Theorem to find the length of the height:

$(5\ m)^2 + B^2 = (13\ m)^2$

$B = 12\ m$

Plugging in our values, we get:

$A = (12\ m)(12\ m)$

$A = 144\ m^2$

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Question

Find 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 is the base of the trapezoid and is the height of the trapezoid.

Use the formula for a triangle to find the length of the base and height:

$a-2a-a\sqrt{3}$

$5\ m-10\ m-5\sqrt{3}\ m$

Use the formula for a triangle to find the length of the base and height:

$5\ m-5\ m-5\sqrt{2}\ m$

Plugging in our values, we get:

$A = \frac{1}{2}(10\ m+5\ m+10\ m+5\sqrt{3}\ m)(5\ m)$

$A = \frac{125+25\sqrt{3}\ m^2}{2}$

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Question

This figure is an isosceles trapezoid with bases of 6 in and 18 in and a side of 10 in.

What is the perimeter of the isoceles trapezoid (in.)?

Answer

The perimeter of the isoceles trapezoid is the sum of all the sides. You can assume the left side is also 10 in. because it is an isoceles trapezoid.

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Question

Find the perimeter of the following trapezoid:

Answer

The formula for the perimeter of a trapezoid is:

$P= b_{1} +b_{2}+e_{1}+e_{2}$

Where is the base and is the edge

Plugging in our values, we get:

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Question

Find the perimeter of the following trapezoid:

Answer

Use the formula for triangles in order to find the lengths of all the sides and bases.

The formula is:

$a-a\sqrt{3}-2a$

Where is the length of the side opposite the $\measuredangle 30$ .

Beginning with the side, if we were to create a triangle, the length of the base is $6\sqrt{3}m$ , and the height is .

Creating another triangle on the left, we find the height is , the length of the base is $2\sqrt{3}m$ , and the side is $4\sqrt{3}m$ .

The formula for the perimeter of a trapezoid is:

$P= b_{1} +b_{2}+e_{1}+e_{2}$

Where is the base and is the edge

Plugging in our values, we get:

$P= (16m) +(16m+6\sqrt{3}m+2\sqrt{3}m)+(12m)+(4\sqrt{3}m)$

$P=44+12\sqrt{3}m$

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Question

Determine the perimeter of the following trapezoid:

Answer

The formula for the perimeter of a trapezoid is:

,

where is the length of the base and is the length of the edge.

Plugging in our values, we get:

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Question

Find the perimeter of the following trapezoid:

Answer

The formula for the perimeter of a trapezoid is:

,

where is the length of the base and is the length of the edge.

Plugging in our values, we get:

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Question

Find the perimeter of the following trapezoid:

Answer

The formula for the perimeter of a trapezoid is

.

Use the formula for a triangle to find the length of the base and side:

$a-2a-a\sqrt{3}$

$5\ m-10\ m-5\sqrt{3}\ m$

Use the formula for a triangle to find the length of the base and side:

$5\ m-5\ m-5\sqrt{2}\ m$

Plugging in our values, we get:

$P = 5\sqrt{2}\ m+10\ m+10\ m+5\sqrt{3}\ m+10\ m+5\ m$

$P = 35+5\sqrt{2}+5\sqrt{3}\ m$

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Question

What is the area of a parallelogram with a base of and a height of ?

Answer

To solve this question you must know the formula for the area of a parallelogram.

In this equation, is the length of the base and is the length of the height. We can plug in the side length for both base and height, as given in the question.

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Question

What is the area of a parallelogram with a base of and a height of ?

Answer

To solve this question you must know the formula for the area of a parallelogram.

The formula is

So we can plug in the side length for both base and height to yield

Perform the multiplication to arrive at the answer of .

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Question

Find the area of the following parallelogram:

Answer

The formula for the area of a parallelogram is:

,

where is the length of the base and is the length of the height.

In order to the find the height of the parallelogram, use the formula for a triangle:

$a-a\sqrt{3}-2a$ , where is the side opposite the $\measuredangle30$ .

The left side of the parallelogram forms the following triangle:

$4m-4\sqrt{3}m-8m$ , where $4\sqrt{3}m$ is the length of the height.

Plugging in our values, we get:

$A=(12m)(4\sqrt{3}m)=48\sqrt{3}m^2$

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Question

Find the area of the following parallelogram:

Answer

Use the Pythagorean Theorem to determine the length of the diagonal:

$B = 5 \sqrt{7}m$

The area of the parallelogram is twice the area of the right triangles:

$A_{Triangle} = \frac{1}{2} (b)(h)$

$A_{Triangle} = \frac{1}{2} (9m)(5m\sqrt{7}) = \frac{45m^2\sqrt{7}}{2}$

$A_{Parallelogram} = 2 (A_{Triangle}) = 2\left(\frac{45m^2\sqrt{7}}{2}\right)=45m^2\sqrt{7}$

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