2-Dimensional Geometry

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GED Math › 2-Dimensional Geometry

Questions 1 - 10

Give the area of a circle with circumference $22 \pi$ .

$121 \pi$

$81 \pi$

$162 \pi$

$242 \pi$

Explanation

The radius of a circle can be determined by dividing its circumference by $2 \pi$ :

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

Set $C = 22\pi$ and evaluate :

$r = \frac{22 \pi}{2 \pi } = 11$

Now substitute 11 for in the area formula for a circle:

$A = \pi r^{2}$

$A = \pi \cdot 11^{2 }$

$A = 121 \pi$ square untis.

A rectangle has length 10 inches and width 5 inches. Each dimension is increased by 3 inches. By what percent has the area of the rectangle increased?

Explanation

The area of a rectangle is its length times its width.

Its original area is $10 \times 5 = 50$ square inches; its new area is $13 \times 8 = 104$ square inches. The area has increased by

$\frac{104 - 50}{50 } \times 100 = \frac{54}{50 } \times 100 = 108 %$ .

A square and a circle share a center as shown by the figure below.

If the length of a side of the square is and is half the length of the radius of the circle, to the nearest hundredths place, find the area of the shaded region.

Explanation

In order to find the area of the shaded region, we will need to find the area of the circle then subtract the area of the square from it.

Start by finding the area of the square. Since we know the side length of the square, we can write the following:

$\text{Area of Square}=4^2=16$

Next, we find the area of the circle. Since the question states that the length of a side of the square is half the radius of the circle, the radius of the circle must be . Thus, we can find the area of the circle.

$\text{Area of Circle}=\pi r^2=\pi(8)^2=64\pi$

Now, subtract these two values to find the area of the shaded region.

$\text{Area of Shaded Region}=64\pi-16=185.06$

Determine the area of the triangle if the base is 12 and the height is 20.

Explanation

Write the formula for the area of a triangle.

$A=\frac{BH}{2}$

Substitute the base and height into the equation.

$A=\frac{(12)(20)}{2} = 6(20) =120$

The answer is:

In the figure above, $a\parallel b$ . If the measure of $\measuredangle 1=6x^{\circ}$ and $\measuredangle 6=4x^{\circ}$ , what is the measure of $\measuredangle8$ ?

$108^{\circ}$

$122^{\circ}$

$72^{\circ}$

$58^{\circ}$

Explanation

Since we have two parallel lines, we know that $\measuredangle1=\measuredangle4$ since they are opposite angle.

We also know that $\measuredangle 4 \text{ and }\measuredangle 6$ are supplementary because they are consecutive interior angles. Thus, we know that $\measuredangle 1$ is also supplementary to $\measuredangle6$ .

We can then set up the following equation to solve for .

Thus, $\measuredangle 1=108^{\circ}$ and $\measuredangle6=72^{\circ}$ .

Now, notice that $\measuredangle 8=\measuredangle 4$ because they are corresponding angles. Thus, $\measuredangle 8=\measuredangle 1=108^{\circ}$ .

The image is not to scale.

If the sum of two angles results in a complementary angle, what is the measure of the unknown angle?

Capture2

$60^{\circ}$

$150^{\circ}$

$30^{\circ}$

$90^{\circ}$

$20^{\circ}$

Explanation

With the provided image, we are asked to solve for the measure of the unknown angle.

Capture2

First, we must understand some information before attempting to solve the problem. The problem provides the information that the two angles summed up result in a complimentary angle. This is another way to say that when we add the measures of the two angles, it will equal $90^{\circ}$ .

This becomes a problem where we solve for a missing variable now. We can call the unknown angle x. We would set this up in equation format accordingly:

Now, we can solve for x.

$x+30{\color{Red} -30}=90{\color{Red} -30}$

$x=60^{\circ}$

Therefore, the unknown angle is $60^{\circ}$ .

Two angles are complementary if they add up to:

$90^{\circ}$

$180^{\circ}$

$270^{\circ}$

$360^{\circ}$

$45^{\circ}$

Explanation

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

If two angles are complementary, and one angle is $75^{\circ}$ , what is the value of the other angle?

$15^{\circ}$

$105^{\circ}$

$30^{\circ}$

$285^{\circ}$

$150^{\circ}$

Explanation

Two angles are complementary if they add up to $90^{\circ}$ . We can use the formula:

$x+y=90^{\circ}$

where x and y are the angles.

Now, we know one angle is $75^{\circ}$ . So, we will substitute and solve for the other angle. We get

$x+75^{\circ}=90^{\circ}$

$x+75^{\circ}-75^{\circ}=90^{\circ}-75^{\circ}$

$x+0^{\circ} = 15^{\circ}$

$x=15^{\circ}$

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

$m\angle A=75^{o}$

$m\angle B=15^{o}$

$m\angle B=25^{o}$

$m\angle B=105^{o}$

$m\angle B=72^{o}$

Explanation

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:

$m\angle B=90^{o}-75^{o}=15^{o}$