2-Dimensional Geometry - GED Math

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Question

Refer to the above diagram. $\Delta ABC$ is equilateral; $\Delta ABX \cong \Delta ACX$ . How many of the following statements must be true?

I) $\overline{AX}$ bisects $\angle BAC$

II) $\overline{AX}$ bisects $\overline{BC}$

III) $\overline{AX} \perp \overline{BC}$

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Answer

Since corresponding parts of congruent triangles are congruent, it follows that $\angle BAX \cong \angle CAX$ . Therefore, $\overline{AX}$ , by definition, bisects $\angle BAC$ , and the first statement is true. A bisector of an angle of an equilateral triangle is also the perpendicular bisector of the opposite side, so the other two statements are immediate consequences.

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Question

Refer to the above diagram. Which of the following expressions gives the length of $\overline{AC}$ ?

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Answer

By the Pythagorean Theorem,

$AC = \sqrt{(AB)^{2} + (BC)^{2}}$

$AC = \sqrt{(x+10)^{2} + x^{2}}$

$AC = \sqrt{ x^{2}+ 2 \cdot x \cdot 10 +10^{2} + x^{2}}$

$AC = \sqrt{ x^{2}+ 2 \cdot x \cdot 10 +10^{2} + x^{2}}$

$AC = \sqrt{ x^{2}+20x +100+ x^{2}}$

$AC = \sqrt{ 2x^{2}+20x +100 }$

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Question

A circular swimming pool at an apartment complex has diameter 50 feet. The apartment manager needs to purchase a tarp that will cover this pool completely, but the store will only sell the material in multiples of ten square yards. How many square yards will the manager need to buy?

Use 3.14 for $\pi$ .

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Answer

The radius of the swimming pool is half the diameter, or 25 feet.

The area of the swimming pool is $\pi$ times the square of the radius, or

$\pi \times 25 ^{2} = 3.14 \times 625 = 1,962.5$ square feet, or

$1,962.5 \div 9 \approx 218$ square yards.

The manager will need to buy a number of square yards of tarp equal to the next highest multiple of ten, which is 220 square yards.

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Question

A circular swimming pool at an apartment complex has diameter 30 meters. The apartment manager needs to purchase a tarp that will cover this pool completely, but the store will only sell the material in multiples of ten square meters. How many square yards will the manager need to buy?

Use 3.14 for $\pi$ .

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Answer

The radius of the swimming pool is half the diameter, or 15 meters.

The area of the swimming pool is $\pi$ times the square of the radius, or

$\pi \times 15 ^{2} = 3.14 \times 225 = 706.5$ square meters.

The manager will need to buy a number of square yards of tarp equal to the next highest multiple of ten, which is 710 square meters.

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Question

What is the area of the circle with a radius of 25?

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Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius into the equation.

$A=\pi (25)^2$

Simplify the equation.

$A=\pi (25)(25) = 625 \pi$

The answer is: $625 \pi$

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Question

Find the area of a circle with a radius of 12.

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Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius into the equation.

$A=\pi (12)^2 = 144\pi$

The answer is: $144\pi$

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Question

What is the area of a circle with the diameter of 18?

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Answer

Write the formula for the area of a circle.

$A=\pi r^2$

The radius is half the diameter, or 9.

Substitute the value of the radius in to the formula.

$A=\pi( 9)^2 = 81\pi$

The answer is: $81\pi$

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Question

Find the area of a circle with a radius of $2\pi$ .

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Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius into the equation.

$A=\pi (2\pi)^2$

Simplify this equation.

$A=\pi (2\pi)(2\pi) = 4\pi ^3$

The answer is: $4\pi ^3$

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Question

Find the area of a circle with a circumference of $100 \pi$ .

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Answer

Write the formula for the circumference of a circle.

$C=2\pi r$

Substitute the circumference.

$100\pi=2\pi r$

Divide by $2\pi$ on both sides.

$\frac{100\pi}{2\pi}=\frac{2\pi r}{2\pi}$

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius into the equation.

$A=\pi (50)^2 = 2500\pi$

The answer is: $2500\pi$

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Question

Find the area of a circle with a diameter of 22cm.

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Answer

To find the area of a circle, we will use the following formula:

$A = \pi \cdot r^2$

where r is the radius of the circle.

Now, we know the diameter of the circle is 22cm. We also know the diameter is two times the radius. Therefore, the radius is 11cm.

So, we can substitute. We get

$A = \pi \cdot (11\text{cm})^2$

$A = \pi \cdot 121\text{cm}^2$

$A = 121\pi \text{ cm}^2$

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Question

Find the area of a circle with a radius of 8in.

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Answer

To find the area of a circle, we will use the following formula:

$A = \pi \cdot r^2$

where r is the radius of the circle.

Now, we know the radius of the circle is 8in. So, we can substitute. We get

$A = \pi \cdot (8\text{in})^2$

$A = \pi \cdot 64\text{in}^2$

$A = 64\pi \text{ in}^2$

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Question

Determine the area of a circle in square feet with a radius of 12 inches.

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Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Convert the radius to feet. There are 12 inches in a foot.

This means the radius in feet is 1.

Substitute the radius in feet to obtain the area in feet squared.

$A=\pi (1\textup{ ft})^2 = \pi \textup{ ft}^2$

The answer is: $\pi$

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Question

Determine the area of a circle if the radius is .

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Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius into the equation.

$A=\pi(20)^2 = 400\pi$

The answer is: $400\pi$

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Question

What is the area of the circle with a diameter of ?

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Answer

Write the formula for the area of a circle.

$A= \pi r^2$

The radius is half the diameter.

$r= \frac{d}{2} = \frac{12}{2} = 6$

Substitute the radius into the equation.

$A= \pi( 6)^2 = 36\pi$

The answer is: $36\pi$

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Question

Find the area of a circle if the circumference is $16\pi^4$ .

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Answer

Write the formula for the circumference of a circle.

$C=2\pi r$

Substitute the circumference.

$16\pi^4 = 2\pi r$

Divide by $2\pi$ on both sides.

$\frac{16\pi^4 }{2\pi}= \frac{2\pi r}{2\pi}$

$r=8\pi ^3$

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius.

$A=\pi (8\pi ^3)^2 = \pi(8\pi ^3)(8\pi ^3) =64\pi ^7$

The answer is: $64\pi ^7$

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Question

Determine the area of a circle with a radius of 15.

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Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius into the equation.

$A=\pi (15)^2 = 225\pi$

The answer is: $225\pi$

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Question

Determine the area of a circle if the radius is $10\pi^2$ .

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Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius into the equation.

$A=\pi (10\pi^2)^2 = 100\pi ^5$

The answer is: $100\pi ^5$

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Question

The figure above is a circle.

If the diameter of the circle above is 10 in., what is the area of the sector? Round to the nearest hundredth.

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Answer

To find the area of a sector, you are looking for a percentage of the total circle's area. Thus, if the angle is degrees, you are looking for $\frac{77}{360}$ of a complete circle.

Given that the diameter is in., you know that the radius is in. This means that the area must be:

$A=\pi r^2 = \pi5^2=25\pi$

The area of the sector is:

$\frac{77}{360}*25\pi=16.79879405044542...$

or

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Question

Find the area of a circle with a diameter of 26cm.

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Answer

To find the area of a circle, we will use the following formula:

$A = \pi \cdot r^2$

where r is the radius of the circle.

Now, we know the diameter of the circle is 26cm. We also know the diameter is two times the radius. Therefore, the radius is 13cm.

So, we can substitute. We get

$A = \pi \cdot (13\text{cm})^2$

$A = \pi \cdot 169\text{cm}^2$

$A = 169\pi \text{ cm}^2$

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Question

Find the area of a circle with a radius of 12in.

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Answer

To find the area of a circle, we will use the following formula:

$A = \pi \cdot r^2$

where r is the radius of the circle.

Now, we know the radius of the circle is 12in. So, we can substitute. We get

$A = \pi \cdot (12\text{in})^2$

$A = \pi \cdot 144\text{in}^2$

$A = 144\pi \text{ in}^2$

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