Geometry - SSAT Upper Level Quantitative

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Question

What is the slope of the given linear equation?

2x + 4y = -7

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Answer

We can convert the given equation into slope-intercept form, y=mx+b, where m is the slope. We get y = (-1/2)x + (-7/2)

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Question

Find the slope of the line 6X – 2Y = 14

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Answer

Put the equation in slope-intercept form:

y = mx + b

-2y = -6x +14

y = 3x – 7

The slope of the line is represented by M; therefore the slope of the line is 3.

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Question

Find the area of a circle with a radius of 5.

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Answer

The formula for the area of a circle is as follows:

$A=\Pi\cdot r^{2}$

In this formula, A is area, and r is for radius. We know the radius of the circle is 5, so plug in the numbers to get the answer. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$A=\Pi\cdot 5^{2}=25\Pi$

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Question

Can we determine the volume of this shape?

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Answer

Volume is a characteristic of three-dimensional shapes. This is a cube, which is a three-dimensional shape. We can think of volume as space within an object or shape, so the volume would be the amount of space inside this cube.

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Question

Find the area of a circle with a diameter of 6.

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Answer

The formula for the area of a circle is as follows:

$A=\Pi\cdot r^{2}$

In this formula, A is area, and r is for radius. We know the diameter of the circle is 6, meaning we have to first find the radius. The diameter is twice the length of the radius of a circle, meaning to find radius divide the diameter by 2:

$\frac{6}{2}=3=r$

Now plug in the numbers to get the answer. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$A=\Pi\cdot 3^{2}=9\Pi$

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Question

What is the slope of the line:

$\frac{14}{3}x=\frac{1}{6}y-7$

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Answer

First put the question in slope intercept form (y = mx + b):

–(1/6)y = –(14/3)x – 7 =>

y = 6(14/3)x – 7

y = 28x – 7.

The slope is 28.

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Question

Find the area of a circle with a diameter of 16.

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Answer

The formula for the area of a circle is as follows:

$A=\Pi\cdot r^{2}$

In this formula, A is area, and r is for radius. We know the diameter of the circle is 16, meaning we have to first find the radius. The diameter is twice the length of the radius of a circle, meaning to find radius divide the diameter by 2:

$\frac{16}{2}=8=r$

Now plug in the numbers to get the answer. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$A=\Pi\cdot 8^{2}=64\Pi$

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Question

If 2x – 4y = 10, what is the slope of the line?

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Answer

First put the equation into slope-intercept form, solving for y: 2x – 4y = 10 → –4y = –2x + 10 → y = 1/2*x – 5/2. So the slope is 1/2.

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Question

What is the slope of the line with equation 4_x_ – 16_y_ = 24?

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Answer

The equation of a line is:

y = mx + b, where m is the slope

4_x_ – 16_y_ = 24

–16_y_ = –4_x_ + 24

y = (–4_x_)/(–16) + 24/(–16)

y = (1/4)x – 1.5

Slope = 1/4

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Question

The perimeter of an equilateral triangle is 39 meters. In meters, how long is one side of the triangle?

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Answer

An equilateral triangle has sides that are the same length. Then, to find the length of one side, you only need to divide the perimeter by 3.

$39\div3=13$

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Question

A circle on the coordinate plane has equation

$x^{2} + y^{2} = 75$ .

Which of the following gives the circumference of the circle?

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Answer

The equation of a circle on the coordinate plane is

$x^{2} + y^{2} = r^{2}$ ,

where is the radius. Therefore,

$r ^{2} = 75$

and

$r = \sqrt{ 75} = \sqrt{25} \times \sqrt{ 3} = 5 \sqrt{ 3}$ .

The circumference of a circle is $2 \pi$ times is radius, which here would be

$C = 2 \pi \times 5 \sqrt{3} = 10\pi \sqrt{3}$ .

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Question

What is the slope of a line that is parallel to the line 11x + 4y - 2 = 9 – 4x ?

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Answer

We rearrange the line to express it in slope intercept form.

Any line parallel to this original line will have the same slope.

$\newline 11x+4y-2=9-4x \newline 4y=11-15x \newline y=\frac{11}{4}-\frac{15}{4}x \newline y=-\frac{15}{4}x+\frac{11}{4}$

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Question

A circle on the coordinate plane has equation

$x^{2} + y^{2} = 49$

Which of the following gives the circumference of the circle?

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Answer

The equation of a circle on the coordinate plane is

$x^{2} + y^{2} = r^{2}$

where is the radius. Therefore,

$r^{2}= 49$

and

$r = \sqrt{49} = 7$ .

The circumference of a circle is $2 \pi$ times is radius, which here would be

$C= 2 \pi \cdot 7 = 14 \pi$

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Question

Refer to the above diagram. Give the length of arc $\widehat{AB}$ .

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Answer

The figure is a sector of a circle with radius 8; the sector has degree measure $180 ^{\circ } - 45 ^{\circ }= 135 ^{\circ }$ . The length of the arc is

$A = \frac{ 135^{\circ }}{360 ^{\circ }} \cdot 2\pi r$

$A = \frac{ 135^{\circ }}{360 ^{\circ }} \cdot 2\pi \cdot 8$

$A = \frac{3}{8} \cdot 16\pi$

$A = 6\pi$

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Question

Find the area of a circle with a radius of 4.

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Answer

The formula for the area of a circle is as follows:

$A=\Pi\cdot r^{2}$

In this formula, A is area, and r is for radius. We know the radius of the circle is 4, so plug in the numbers to get the answer. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$A=\Pi\cdot 4^{2}=16\Pi$

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Question

A $60 ^{\circ }$ central angle of a circle has a chord with length 9. Give the circumference of the circle.

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Answer

The figure below shows $\angle AOB$ , which matches this description, along with its chord $\overline{AB}$ :

By way of the Isosceles Triangle Theorem, $\Delta AOB$ can be proved equilateral, so . This is the radius, so the circumference is

$C = 2 \pi r = 2 \pi \cdot 9 = 18 \pi$

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Question

Find the slope of the line that passes through the points $(2, -3)\text{ and }(-3, -3)$

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Answer

Use the following formula to find the slope:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{-3-(-3)}{-3-2}=\frac{0}{-5}=0$

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Question

Given the graph of the line below, find the equation of the line.

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Answer

To solve this question, you could use two points such as (1.2,0) and (0,-4) to calculate the slope which is 10/3 and then read the y-intercept off the graph, which is -4.

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Question

Find the length of a line segment that has endpoints at $(1, 2)\text{ and }(5, 1)$ .

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Answer

Use the distance formula to find the length of the line segment.

$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Where,

and

$\text{Distance}=\sqrt{(5-1)^2+(1-2)^2}=\sqrt{16+1}=\sqrt{17}$

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Question

Define an operation $\wedge$ as follows:

For all complex numbers ,

$a \wedge b = a \div (b+ 2i)$

Evaluate $8 \wedge 4$

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Answer

$a \wedge b = a \div (b+ 2i) = \frac{a}{b+2i}$

$8 \wedge 4 = \frac{8}{4+2i}$

Multiply both numerator and denominator by the conjugate of the denominator, , to rationalize the denominator:

$8 \wedge 4 = \frac{8(4-2i)}{(4+2i)(4-2i)}$

$= \frac{8\cdot 4-8\cdot 2i}{4^{2}+2^{2}}$

$= \frac{32-16i}{16+4}$

$= \frac{32-16i}{20}$

$= \frac{32 }{20} - \frac{ 16}{20}i$

$= \frac{8}{5} -\frac{ 4}{5}i$

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