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HSPT Math : Geometry

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Example Questions

Example Question #13 : How To Find The Measure Of An Angle

Spinner target 2

The above diagram shows a spinner. The radius of the smaller quarter-circles is half that of the larger quarter-circles.

A player spins the above spinner. What are the odds against the spinner landing while pointing inside one of the blue regions?

Possible Answers:

17 to 7

13 to 11

8 to 5

5 to 2

Correct answer:

17 to 7

Explanation:

The size of the regions does not matter here. What matters is the angle measurement, or, equivalently, what part of a circle each sector is.

Two of the blue sectors are each one third of one quarter-circle, and thus are

$\frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$

of one circle.

The other two blue sectors are each one fourth of one quarter-circle, and thus are

$\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$

of one circle.

Therefore, the total angle measure comprises

$\frac{1}{16} + \frac{1}{16} + \frac{1}{12} + \frac{1}{12}$

$=\frac{3}{48} + \frac{3}{48} + \frac{4}{48} + \frac{4}{48}$

$= \frac{14}{48} = \frac{7}{24}$

of a circle. This makes $\frac{7}{24}$ the correct probability. As odds, this translates to

(24 -7): 7 , or 17:7 odds against the spinner landing in blue.

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Example Question #14 : How To Find The Measure Of An Angle

Spinner target 2

The above diagram shows a spinner. The radius of the smaller quarter-circles is half that of the larger quarter-circles.

A player spins the above spinner. What are the odds against the spinner landing while pointing inside the purple region?

Possible Answers:

11 to 1

9 to 1

13 to 1

7 to 1

Correct answer:

11 to 1

Explanation:

The size of the regions does not matter here. What matters is the angle measurement, or, equivalently, what part of a circle each sector is.

The purple region is one third of one quarter of a circle, or, equvalently,

$\frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$

of a circle, so its central angle is $\frac{1}{12}$ of the total measures of the angles of the sectors. This makes $\frac{1}{12}$ the probability of the spinner stopping inside the purple region; this translates to

(12-1):1 or 11:1 odds against this occurrence.

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Example Question #15 : How To Find The Measure Of An Angle

In parallelogram ABCD , $m \angle A = (t+ 56 )^{\circ }$ . Give the measure of $\angle B$ in terms of .

Possible Answers:

$(124-t )^{\circ }$

$(t+ 56 )^{\circ }$

$(34-t )^{\circ }$

$(t+ 146)^{\circ }$

Correct answer:

$(124-t )^{\circ }$

Explanation:

$\angle A$ and $\angle B$ are a pair of adjacent angles of the parallelogram, and as such, they are supplementary - that is, their degree measures total 180. Therefore,

$m \angle A + m \angle B = 180 ^{\circ }$

$m \angle B = 180 ^{\circ } - m \angle A$

$= 180 ^{\circ } - (t+ 56 )^{\circ }$

$= (180 - 56 - t )^{\circ }$

$= (124-t )^{\circ }$

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Example Question #16 : How To Find The Measure Of An Angle

The measures of the angles of $\bigtriangleup ABC$ are as follows:

$m \angle A = (3t-14)^{\circ }$

$m \angle B = (2t+16)^{\circ }$

$m \angle C =( 5t)^{\circ }$

Is this triangle acute, obtuse, right, or nonexistent?

Possible Answers:

$\bigtriangleup ABC$ is a right triangle

$\bigtriangleup ABC$ cannot exist

$\bigtriangleup ABC$ is an acute triangle

$\bigtriangleup ABC$ is an obtuse triangle

Correct answer:

$\bigtriangleup ABC$ is an acute triangle

Explanation:

The sum of the measures of the angles of a triangle is 180 degrees, so solve for in the equation:

$m \angle A + m \angle B + m \angle C = 180^{\circ }$

(3t-14) + (2t+16) + 5t = 180

10t+ 2 = 180

10t+ 2- 2 = 180 - 2

10t = 178

$10t \div 10 = 178 \div 10$

t = 17.8

$m \angle A = (3t-14)^{\circ } = (3 \cdot 17.8-14)^{\circ } = (53.4-14)^{\circ } = 39.4^{\circ }$

$m \angle B = (2t+16)^{\circ } = (2 \cdot 17.8 + 16)^{\circ }= (35.6+ 16) ^{\circ }= 51.6^{\circ }$

$m \angle C =( 5t)^{\circ } = ( 5 \cdot 17.8)^{\circ } = 89 ^{\circ }$

All three angles measure less than 90 degrees and are therefore acute angles; that makes $\bigtriangleup ABC$ an acute triangle.

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Example Question #12 : How To Find The Measure Of An Angle

If you have a right triangle, what is the measure of the two of the angles if they are equal?

Possible Answers:

135

Correct answer:

Explanation:

The total degrees of the angles in a triangle are 180 .

Since it is a right triangle, one of the three angles must be 90.

That leaves you with for the other two angles (180-90=90) .

If they are equal, you just divide the remaining degrees by to get 90/2=45 .

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Example Question #11 : How To Find The Measure Of An Angle

If you have a right triangle with an angle measuring 45 degrees, what is the third angle measurement?

Possible Answers:

$180^\circ$

$45^\circ$

$90^\circ$

$60^\circ$

$30^\circ$

Correct answer:

$45^\circ$

Explanation:

A right triangle has one 90 degree angle and all three angles must equal 180 degrees.

To find the answer, just subtract the two angles you have from the total to get

$180^\circ-\textup{Angle 1}-\textup{Angle 2}=\textup{Angle 3}$

The angle we have are,

$\\ \textup{Angle 1}=90^\circ \\ \textup{Angle 2}=45^\circ$ .

Substituting these into the formula results in the solution.

$180^\circ-90^\circ-45^\circ=45^\circ$ .

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Example Question #1 : How To Do Coordinate Geometry

A deer walks in a straight line for 8 hours. At the end of its journey, the deer is 30 miles north and 40 miles east of where it began. What was the average speed of the deer?

Possible Answers:

12.5 miles per hour

6.25 miles per hour

miles per hour

Correct answer:

6.25 miles per hour

Explanation:

To find the speed of the deer, you must have the distance traveled and the time.

The distance is found using the Pythagorean Theorem:

A^2 + B^2 = C^2

$C = {\sqrt{A^2+ B^2}}$

$C = \sqrt{30^2 + 40^2}$

$C = \sqrt{2500} = 50$

The answer must be in miles per hour so the total miles are divided by the hours to get the final answer:

$\frac{miles}{hour} = \frac{50}{8} = 6.25 mph$

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Example Question #683 : Sat Mathematics

You are looking at a map of your town and your house is located at the coordinate (0,0). Your school is located at the point (3,4). If each coordinate distance is 1.3 miles, how far away is your school?

Possible Answers:

$6.5\ miles$

$9.1\ miles$

$5\ miles$

$32.5\ miles$

$7\ miles$

Correct answer:

$6.5\ miles$

Explanation:

The coordinate length between you and your school is equivalent to the hypotenuse of a right triangle with sides of 3 and 4 units:

x^2=3^2+4^2

x^2=9+16

x^2=25

x=5

The distance is 5 coordinate lengths, and each coordinate length corresponds to 1.3 miles of distance, so

$5\times 1.3 =6.5\hspace{1 mm}miles$

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Example Question #1 : How To Do Coordinate Geometry

Which of the following is a vertex of the square?

Question_12

Possible Answers:

$\small (-2,1)$

$\small (2,4)$

$\small (-1,2)$

$\small (-1,-4)$

$\small (1,-4)$

Correct answer:

$\small (-1,2)$

Explanation:

The coordinates of a point are determined by the distance from the origin. The first point in the ordered pair is the number of units to the left or right of the origin. Negative numbers indicate the number of units to the left while positive numbers indicate the number of units to the right. The second number indicates the number of units above or below the origin. Positive numbers indicate the number of units above while negative numbrs indicate the number of units below the origin. The vertices of the square are:
$\small (-1,2); (-1,4); (1,2); (-1,4)$

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Example Question #1 : Coordinate Geometry

Which of the following points will you find on the -axis?

Possible Answers:

(30,30)

(-30,30)

(0,-30)

(30, 0)

(30,-30)

Correct answer:

(0,-30)

Explanation:

A point is located on the -axis if and only if it has -coordinate (first coordinate) 0. Of the five choices, only (0,-30) fits that description.

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HSPT Math : Geometry

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #13 : How To Find The Measure Of An Angle

Example Question #14 : How To Find The Measure Of An Angle

Example Question #15 : How To Find The Measure Of An Angle

Example Question #16 : How To Find The Measure Of An Angle

Example Question #12 : How To Find The Measure Of An Angle

Example Question #11 : How To Find The Measure Of An Angle

Example Question #1 : How To Do Coordinate Geometry

Example Question #683 : Sat Mathematics

Example Question #1 : How To Do Coordinate Geometry

Example Question #1 : Coordinate Geometry

All HSPT Math Resources

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