HSPT Math : Geometry

Study concepts, example questions & explanations for HSPT Math

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Example Questions

Example Question #221 : Geometry

A sphere has diameter 12. What is 75% of its volume?

Possible Answers:

\displaystyle 108 \pi

\displaystyle 432 \pi

\displaystyle 216 \pi

\displaystyle 1,728 \pi

Correct answer:

\displaystyle 216 \pi

Explanation:

The radius of a sphere is half its diameter, which here is 12, so the radius is 6. The volume of the sphere can be calculated by setting \displaystyle r = 6 in the formula:

\displaystyle V = \frac{4}{3} \pi r ^{3}

\displaystyle V = \frac{4}{3} \pi \cdot 6 ^{3} = \frac{4}{3} \pi \cdot 216 = 288 \pi

75% of this is

\displaystyle 288 \pi \cdot \frac{75}{100} = 216 \pi

Example Question #222 : Geometry

Find the volume of a rectangular prism with the following information:

\displaystyle \\height=5 \\length=4 \\width=3

 

Possible Answers:

\displaystyle \frac{20}{3}

\displaystyle 60

\displaystyle 20

\displaystyle 12

Correct answer:

\displaystyle 60

Explanation:

The formula for volume of a rectangular prism is,

 \displaystyle V=length*width*height.  

For this problem, since

\displaystyle \\height=5 \\length=4 \\width=3

the solution would be 

\displaystyle V=5*4*3=60.

Example Question #223 : Geometry

What is the measure of a right angle?

Possible Answers:

\displaystyle 45

\displaystyle 180

\displaystyle 360

\displaystyle 90

Correct answer:

\displaystyle 90

Explanation:

The measure of a right angle is \displaystyle 90 degrees.

Example Question #224 : Geometry

Triangle_a

Figure NOT drawn to scale.

If \displaystyle w = 125^{\circ } and \displaystyle x = 81^{\circ }, evaluate \displaystyle y.

Possible Answers:

More information is needed to solve the problem.

\displaystyle 55^{\circ }

\displaystyle 53^{\circ }

\displaystyle 44^{\circ }

\displaystyle 35^{\circ}

Correct answer:

\displaystyle 44^{\circ }

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

\displaystyle w = x + y

\displaystyle y = w - x= 125 - 81 = 44^{\circ }

Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

Triangle_a

Figure NOT drawn to scale.

If \displaystyle x = 72^{\circ } and \displaystyle y = 43^{\circ }, evaluate \displaystyle w.

Possible Answers:

\displaystyle 65^{\circ}

\displaystyle 108^{\circ}

\displaystyle 125^{\circ }

\displaystyle 137^{\circ}

\displaystyle 115^{\circ }

Correct answer:

\displaystyle 115^{\circ }

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

\displaystyle w = x + y = 72 + 43 = 115 ^{\circ }

Example Question #225 : Geometry

If the vertex angle of an isoceles triangle is \displaystyle 64^{\circ}, what is the value of one of its base angles?

Possible Answers:

\displaystyle 64^{\circ}

\displaystyle 116^{\circ}

\displaystyle 36^{\circ}

\displaystyle 58^{\circ}

\displaystyle 26^{\circ}

Correct answer:

\displaystyle 58^{\circ}

Explanation:

In an isosceles triangle, the base angles are the same. Also, the three angles of a triangle add up to \displaystyle 180^{\circ}.

So, subtract the vertex angle from \displaystyle 180^{\circ}. You get \displaystyle 116^{\circ}.

Because there are two base angles you divide \displaystyle 116^{\circ} by \displaystyle 2, and you get \displaystyle 58^{\circ}.

Example Question #1411 : Concepts

Triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. 

\displaystyle 45^{\circ } \leq m \angle 1 \leq 50^{\circ }

\displaystyle 60^{\circ } \leq m \angle 2 \leq 70 ^{\circ }

Which of the following could be a measure of \displaystyle \angle 3 ?

Possible Answers:

\displaystyle 125^{\circ }

\displaystyle 100^{\circ }

\displaystyle 130^{\circ }

All of the other choices give a possible measure of \displaystyle \angle 3.

\displaystyle 110^{\circ }

Correct answer:

\displaystyle 110^{\circ }

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

\displaystyle \angle 3 = \angle 1 + \angle 2.

We also have the following constraints:

\displaystyle 45^{\circ } \leq m \angle 1 \leq 50^{\circ }

\displaystyle 60^{\circ } \leq m \angle 2 \leq 70 ^{\circ }

Then, by the addition property of inequalities,

\displaystyle 45^{\circ } + 60 ^{\circ }\leq m \angle 1 +m \angle 2 \leq 50^{\circ } + 70 ^{\circ }

\displaystyle 105 ^{\circ }\leq m \angle 3 \leq 120^{\circ }

Therefore, the measure of \displaystyle \angle 3 must fall in that range. Of the given choices, only \displaystyle 110^{\circ } falls in that range.

Example Question #221 : Geometry

Triangle

Refer to the above diagram. 

\displaystyle 45^{\circ } \leq m \angle 1 \leq 50^{\circ }

\displaystyle 130^{\circ } \leq m \angle 3 \leq 140 ^{\circ }

Which of the following could be a measure of \displaystyle \angle 2 ?

Possible Answers:

\displaystyle 80 ^{\circ }

All of the other responses are correct.

\displaystyle 85 ^{\circ }

\displaystyle 95 ^{\circ }

\displaystyle 90 ^{\circ }

Correct answer:

All of the other responses are correct.

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

\displaystyle \angle 3 = \angle 1 + \angle 2

or 

\displaystyle \angle 2 = \angle 3 - \angle 1

Therefore, the maximum value of \displaystyle \angle 2 is the least possible value of \displaystyle \angle 1 subtracted from the greatest possible value of \displaystyle \angle 3:

\displaystyle 140^{\circ } - 45^{\circ } = 95 ^{\circ }

The minimum value of \displaystyle \angle 2 is the greatest possible value of \displaystyle \angle 1 subtracted from the least possible value of \displaystyle \angle 3:

\displaystyle 130^{\circ } - 50^{\circ } = 80 ^{\circ }

Therefore, 

\displaystyle 80^{\circ } \leq \angle 2 \leq 95^{\circ }

Since all of the choices fall in this range, all are possible measures of \displaystyle \angle 2.

Example Question #1 : How To Find An Angle In A Right Triangle

One angle of a right triangle has measure \displaystyle 120^{\circ }. Give the measures of the other two angles.

Possible Answers:

\displaystyle 90^{\circ }, 120^{\circ }

\displaystyle 30^{\circ }, 90^{\circ }

\displaystyle 30^{\circ }, 30^{\circ }

This triangle cannot exist.

\displaystyle 120^{\circ }, 120^{\circ }

Correct answer:

This triangle cannot exist.

Explanation:

A right triangle must have one right angle and two acute angles; this means that no angle of a right triangle can be obtuse. But since \displaystyle 120^{\circ } > 90^{\circ }, it is obtuse. This makes it impossible for a right triangle to have a \displaystyle 120^{\circ } angle.

Example Question #1 : How To Find An Angle In A Right Triangle

One angle of a right triangle has measure \displaystyle 68^{\circ }. Give the measures of the other two angles.

Possible Answers:

This triangle cannot exist.

\displaystyle 44^{\circ },68^{\circ }

\displaystyle 56^{\circ },56^{\circ }

\displaystyle 22^{\circ }, 90^{\circ }

\displaystyle 52^{\circ }, 60^{\circ }

Correct answer:

\displaystyle 22^{\circ }, 90^{\circ }

Explanation:

One of the angles of a right triangle is by definition a right, or \displaystyle 90^{\circ }, angle, so this is the measure of one of the missing angles. Since the measures of the angles of a triangle total \displaystyle 180^{\circ }, if we let the measure of the third angle be \displaystyle x, then:

\displaystyle x + 68 + 90 = 180

\displaystyle x + 158 = 180

\displaystyle x + 158 - 158= 180 - 158

\displaystyle x = 22

The other two angles measure \displaystyle 22 ^{\circ }, 90^{\circ }.

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