2-Dimensional Geometry - GED Math

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Question

Find the circumference of the circle with a diameter of $2\pi -3$ .

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Answer

Write the formula for the circumference of the circle.

$C=2\pi r = \pi D$

Substitute the diameter.

$C= \pi (2\pi-3) = 2\pi ^2-3\pi$

The answer is: $2\pi ^2-3\pi$

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Question

Determine the circumference of the circle with a diameter of $\frac{5}{3}$ .

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Answer

Write the formula for the circumference of the circle.

$C= 2\pi r = \pi D$

Substitute the diameter.

$C = \pi(\frac{5}{3})$

The answer is: $\frac{5}{3} \pi$

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Question

What is the diameter of a circle with a radius of 9?

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Answer

The diameter of a circle is twice the radius:

Plug in the radius value:

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Question

What is the radius of a circle given the diameter is 22?

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Answer

The radius is half of the diameter, or 11.

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Question

What angle is complementary to 10 degrees?

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Answer

Complementary angles must add up to ninety.

Subtract the given angle from 90 to find the other angle.

The answer is:

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Question

Complementary angles add up to how many degrees?

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Answer

Two angles are complementary when they add up to $90$^{\circ}$$ .

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Question

Give the area of the above circle.

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Answer

The area of a circle, given its radius , can be found using the formula

$A = \pi r^{2}$

The radius is half the diameter - that is, $r = \frac{d}{2}$ - so, substituting,

$A = \pi \left ( \frac{d}{2} \right )^{2}$

or

$A = \pi \left ( \frac{d^{2}}{2^{2}} \right )$

$A = \left ( \frac{d^{2}}{4} \right )\pi$

The diameter of the given circle is 7, so set in the above formula:

$A = \left ( \frac{7^{2}}{4} \right )\pi$

$A = \frac{49}{4} \pi$ ,

the correct area.

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Question

Let $\pi=3.14$ .

Find the area of a circle with a diameter 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 $\pi=3.14$ . We also know the diameter is 12in. We know the diameter is two times the radius, so the radius is 6in. Now, we can substitute. We get

$A = 3.14 \cdot (6\text{in})^2$

$A = 3.14 \cdot 36\text{in}^2$

$A = 113.04\text{in}^2$

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Question

While doing your homework, you become distracted by the 3 holes on the margin of your paper. You estimate that the holes have a diameter of 2.5 cm. What is the circumference of the circles?

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Answer

While doing your homework, you become distracted by the 3 holes on the margin of your paper. You estimate that the holes have a diameter of 2.5 cm. What is the circumference of the circles?

To find circumference, use the following formula:

$C=2 \pi r$

Now, we could find the radius and then plug it in, but if you are astute, you will see that the above formula is the same thing as:

$C= \pi d$

Where d is the diameter.

So, we can plug in our diameter to find our answer:

$C= \pi (2.5cm)\approx 7.85cm$

So we can say our answer is 7.85 cm

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Question

A circle is inscribed in square that has a side length of , as shown by the figure below.

Find the area of the shaded region. Use $\pi=3.14$ .

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Answer

Since the circle is inscribed in the square, the diameter of the circle is the same length as the length of a square.

Start by finding the area of the square.

$\text{Area of Square}=\text{side}^2$

For the given square,

$\text{Area}=144$

Now, because the diameter of the circle is the same as the length of a side of the square, we now also know that the radius of the circle must be . Next recall how to find the area of a circle.

$\text{Area of Circle}=\pi r^2$

Plug in the found radius to find the area of the circle.

$\text{Area of Circle}=\pi(6)^2=36\pi=113.04$

Now, the shaded area is the area left over when the area of the circle is subtracted from the area of the square. Thus, we can write the following equation to find the area of the shaded region.

$\text{Area of Shaded Region}=144-113.04=30.96$

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Question

Give the circumference of the above circle. Assume each mark on each axis represents one unit.

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Answer

The diameter of the circle - the distance from one point to the opposite point - is 10 units, so the circumference is this multiplied by $\pi$ , or $10 \pi$ .

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Question

What is the area of a circle with a diameter of $12\pi$ ?

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Answer

Be careful on several counts for this question. First, remember that you need the radius for calculating area. Therefore, since you know that the diameter is $12\pi$ , you can say that the radius is $6\pi$ .

Next, be very careful given that there is already a $\pi$ in your radius. The formula for the area of a circle is:

$A=\pi r^2$

For your data, this is:

$A=\pi (6\pi)^2=\pi * 36\pi^2=36\pi^3$

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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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