2-Dimensional Geometry - GED Math

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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

Find the the measure of angle B if it is complement of angle A:

$m\angle A=53^{o}$

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Answer

If two angles are complementary, that means the sum of their degrees of measure will add up to 90. In order to find the measure of angle B, subtract angle A from 90 like shown:

$m\angle B=90^{o}-53^{o}=37^{o}$

This gives us a final answer of 37 degrees for angle B.

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Question

Find the the measure of angle B if it is complement of angle A:

$m\angle A=9^{o}$

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Answer

If two angles are complementary, that means the sum of their degrees of measure will add up to 90. In order to find the measure of angle B, subtract angle A from 90 like shown:

$m\angle B=90^{o}-9^{o}=81^{o}$

This gives us a final answer of 81 degrees for angle B.

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Question

If an angle measures degrees, find the measurement of the other angle such that the two angles are complementary.

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Answer

Two angles are classified as complementary if an only if the sum of the angles equals to exactly 90 degrees.

Since we know the measurement of one angle, and we know the rule about complementary angles, let's find the other angle:

The other angle is simply the total sum of both angles minus the given angle.
The total sum of the angles is , and the given angle is .

So, we will subtract from .

Other angle=

The other angle has a measurement of degrees.

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Question

If one angle of an isosceles triangle measures 120, what are the other two angle measures?

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Answer

First we need to recall that whenever we add up all 3 angles of any given triangle, the sum will always be $180^{\circ}$ .

In an isosceles triangle two of the angles are congruent. Since we are told that one of the angles of our triangle is $120^{\circ}$ we know that this is an obtuse triangle, since 120 is greater than 90.

We need to subtract 120 from 180 to find the remainder of the triangle which is $60^{\circ}$

Since we are working with an isosceles triangle, we know that the remaining two angles are going to be congruent. To find the degree of the angles we simply divide 60 by 2. Our answer is; both angles are $30^{\circ}$

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Question

Note: Figure NOT drawn to scale.

Refer to the figure above. Give the area of the blue triangle.

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Answer

The inscribed rectangle is a 20 by 20 square. Since opposite sides of the square are parallel, the corresponding angles of the two smaller right triangles are congruent; therefore, the two triangles are similar and, by definition, their sides are in proportion.

The small top triangle has legs 10 and 20; the blue triangle has legs 20 and , where can be calculated with the following proportion:

$\frac{N}{20} = \frac{20}{10 }$

$10 N = 20 \cdot 20$

$N = 400 \div 10 = 40$

The legs of the blue triangle are 20 and 40; half their product is the area:

$\frac{1}{2} \times 20 \times 40 = 400$

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. In terms of area, $\Delta BDC$ is what percent of $\Delta ABC$ ?

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Answer

The area of a triangle is half the product of its baselength and its height.

To find the area of $\Delta BDC$ , we can use the lengths of the legs $\overline{CD}$ and $\overline{BD}$ :

$A = \frac{1}{2} (CD)(BD ) = \frac{1}{2} \cdot 24 \times 12 = 144$

To find the area of $\Delta ABC$ , we can use the hypotenuse $\overline{AC}$ , the length of which is 30, and the altitude $\overline{BD}$ perpendicular to it:

$A = \frac{1}{2} (AC)(BD ) = \frac{1}{2} \cdot 30 \times 12 = 180$

In terms of area, $\Delta BDC$ is

$\frac{144}{180} \times 100 % = 80 %$

of $\Delta ABC$ .

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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

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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