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HSPT Math : Geometry

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Example Questions

Example Question #2 : Coordinate Geometry

Which of the following points is on the -axis?

Possible Answers:

(20,1)

(20,0)

(0,20)

(10,10)

(0,10)

Correct answer:

(20,0)

Explanation:

A point is located on the -axis if and only if it has a -coordinate equal to zero. So the answer is (20,0) .

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Example Question #4 : How To Do Coordinate Geometry

The point (-1,-1) is reflected across x=1 . What is the new point?

Possible Answers:

(3,-1)

(0,-1)

(4,-1)

(0,1)

(2,-1)

Correct answer:

(3,-1)

Explanation:

The horizontal distance from point (-1,-1) to the vertical line x=1 is two units. Since this point is reflected across x=1 , the new point will also be 2 units to the right of line x=1 .

Therefore, the correct answer is: (3,-1)

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Example Question #1 : How To Do Coordinate Geometry

What is the slope given the following two points? $(2,3) \textup{ and } (3,-8)$

Possible Answers:

$-\frac{1}{11}$

$\frac{1}{5}$

$-\frac{1}{5}$

Correct answer:

Explanation:

Write the slope formula and substitute the two points.

$m=\frac{y_2-y_1}{x_2-x_1}= \frac{-8-3}{3-2}= \frac{-11}{1} = -11$

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Example Question #3 : How To Do Coordinate Geometry

The lines of which two of the following equations are perpendicular to each other?

(I) $y= \frac{6}{11} x$

(II) $y= - \frac{6}{11} x$

(III) $y= - \frac{11}{6} x$

Possible Answers:

No two of the equations given are represented by lines that are perpendicular to each other.

I and II

II and III

I and III

Correct answer:

I and III

Explanation:

All three equations are in slope-intercept form y = mx+ b , so the slope of each line is the coefficient of :

(I) $y= \frac{6}{11} x$ - the slope is $\frac{6}{11}$

(II) $y= - \frac{6}{11} x$ - the slope is $- \frac{6}{11}$

(III) $y= - \frac{11}{6} x$ - the slope is $- \frac{11}{6}$

Two lines are perpendicular if and only if the product of their slopes is .

The product of the slopes of the lines in I and II is

$\frac{6}{11} \times \left (- \frac{6}{11} \right )= - \frac{6 \times 6 }{11 \times 11} =- \frac{36 }{121} \ne -1$

The product of the slopes of the lines in II and III is

$- \frac{6}{11} \times \left (- \frac{11} {6}\right )= + \frac{6 \times 11 }{11 \times 6} = \frac{66}{66} = 1 \ne -1$

The product of the slopes of the lines in I and III is

$\frac{6}{11} \times \left (- \frac{11} {6}\right )= - \frac{6 \times 11 }{11 \times 6} =- \frac{66}{66} = -1$

This makes the lines in I and III perpendicular, and this is the correct choice.

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Example Question #5 : How To Do Coordinate Geometry

Begin at the origin of the rectangular coordinate plane. Move up three units, left seven units, and down nine units. Give the coordinates of your current location.

Possible Answers:

(-7, -6)

( 7, -6)

(-7,6)

( 7,6)

Correct answer:

(-7, -6)

Explanation:

Since you have moved left seven units, you have moved seven units in a negative horizontal direction, making the -coordinate of your current location .

Since you have moved up three units and down nine units, you have moved six units down - this is six units in a negative vertical direction, making the -coordinate of your current location .

Therefore, the ordered pair for your current location is (-7, -6) .

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Example Question #664 : Sat Mathematics

Billy set up a ramp for his toy cars. He did this by taking a wooden plank and putting one end on top of a brick that was 3 inches high. He then put the other end on top of a box that was 9 inches high. The bricks were 18 inches apart. What is the slope of the plank?

Possible Answers:

$\frac{1}{3}$

$\frac{3}{18}$

$\frac{5}{9}$

$\frac{1}{18}$

Correct answer:

$\frac{1}{3}$

Explanation:

The value of the slope (m) is rise over run, and can be calculated with the formula below:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The coordinates of the first end of the plank would be (0,3), given that this is the starting point of the plank (so x would be 0), and y would be 3 since the brick is 3 inches tall.

The coordinates of the second end of the plank would be (18,9) since the plank is 18 inches long (so x would be 18) and y would be 9 since the box was 9 inches tall at the other end.

From this information we know that we can assign the following coordinates for the equation:

$\left(x_{1},y_{1}\right) = (0,3)$ and $(x_{2}, y_{2})= (18,9)$

Below is the solution we would get from plugging this information into the equation for slope:

$m = \frac{9-3}{18-0}}$

This reduces to $\frac{6}{18} =\frac{1}{3}$

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Example Question #231 : Geometry

According to regulations, the maximum radius of a (spherical) bowling ball is 7.11 centimeters. What does that make the maximum surface area, to the nearest square centimeter?

(Note: disregard the holes)

Possible Answers:

$1,506 \; cm^{2}$

$635 \; cm^{2}$

$159 \; cm^{2}$

$376 \; cm^{2}$

$1,129 \; cm^{2}$

Correct answer:

$635 \; cm^{2}$

Explanation:

Set r = 7.11 and use the formula for the surface area of a sphere:

$A = 4\pi r^{2} = 4\pi\cdot 7.11^{2} \approx 635\; cm^{2 }$

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Example Question #1 : Cylinders

An upright cylinder with a height of 30 and a radius of 5 is in a big tub being filled with oil. If only the top 10% of the cylinder is visible, what is the surface area of the submerged cylinder?

Possible Answers:

345π

295π

270π

300π

325π

Correct answer:

295π

Explanation:

The height of the submerged part of the cylinder is 27cm. 2πrh + πr²is equal to 270π + 25π = 295π

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Example Question #1 : How To Find The Surface Area Of A Cylinder

The diameter of the lid of a right cylindrical soup can is 5 in. If the can is 12 inches tall and the label costs $0.00125 per square inch to print, what is the cost to produce a label for a can? (Round to the nearest cent.)

Possible Answers:

$0.16

$0.24

$1.18

$0.08

$0.29

Correct answer:

$0.24

Explanation:

The general mechanics of this problem are simple. The lateral area of a right cylinder (excluding its top and bottom) is equal to the circumference of the top times the height of the cylinder. Therefore, the area of this can's surface is: 5π * 12 or 60π. If the cost per square inch is $0.00125, a single label will cost 0.00125 * 60π or $0.075π or approximately $0.24.

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Example Question #2 : How To Find The Surface Area Of A Cylinder

Aluminum is sold to a soup manufacturer at a rate of $0.0015 per square inch. The cans are made so that the ends perfectly fit on the cylindrical body of the can. It costs $0.00125 to attach the ends to the can. The outer label (not covering the top / bottom) costs $0.0001 per in² to print and stick to the can. The label must be 2 inches longer than circumference of the can. Ignoring any potential waste, what is the manufacturing cost (to the nearest cent) for a can with a radius of 5 inches and a height of 12 inches?

Possible Answers:

$0.84

$0.77

$0.45

$0.57

$0.91

Correct answer:

$0.84

Explanation:

We have the following categories to consider:

<Aluminum Cost> = (<Area of the top and bottom of the can> + <Lateral area of the can>) * 0.0015

<Label Cost> = (<Area of Label>) * 0.0001

<Attachment cost> = 2 * 0.00125 = $0.0025

The area of ends of the can are each equal to π*5² or 25π. For two ends, that is 50π.

The lateral area of the can is equal to the circumference of the top times the height, or 2 * π * r * h = 2 * 5 * 12 * π = 120π.

Therefore, the total surface area of the aluminum can is 120π + 50π = 170π. The cost is 170π * 0.0015 = 0.255π, or approximately $0.80.

The area of the label is NOT the same as the lateral area of the can. (Recall that it must be 2 inches longer than the circumference of the can.) Therefore, the area of the label is (2 + 2 * π * 5) * 12 = (2 + 10π) * 12 = 24 + 120π. Multiply this by 0.0001 to get 0.0024 + 0.012π = (approximately) $0.04.

Therefore, the total cost is approximately 0.80 + 0.04 + 0.0025 = $0.8425, or $0.84.

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HSPT Math : Geometry

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #2 : Coordinate Geometry

Example Question #4 : How To Do Coordinate Geometry

Example Question #1 : How To Do Coordinate Geometry

Example Question #3 : How To Do Coordinate Geometry

Example Question #5 : How To Do Coordinate Geometry

Example Question #664 : Sat Mathematics

Example Question #231 : Geometry

Example Question #1 : Cylinders

Example Question #1 : How To Find The Surface Area Of A Cylinder

Example Question #2 : How To Find The Surface Area Of A Cylinder

All HSPT Math Resources

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