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ISEE Upper Level Math : Plane Geometry

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find The Height Of An Eqilateral Triangle

The length of one side of an equilateral triangle is 4 centimeters. Give the height (altitude) of the triangle.

Possible Answers:

$3\ cm$

$4\ cm$

$2\sqrt{3}\ cm$

$3\sqrt{3}\ cm$

$4\sqrt{3}\ cm$

Correct answer:

$2\sqrt{3}\ cm$

Explanation:

$Area=\frac{1}{2}absinC$ ,

where and are the lengths of two sides and is the corresponding angle.

In an equilateral triangle, all of the sides have the same length, and all three angles are $60^{\circ}$ .

$Area=\frac{1}{2}absinC=\frac{1}{2}\times 4\times 4\times sin60^{\circ}$

$sin60^{\circ}=\frac{\sqrt{3}}{2}\Rightarrow Area=\frac{1}{2}\times 4\times 4\times \frac{\sqrt{3}}{2}=4\sqrt{3}\ cm^2$

Now plug this area into the alternate formula for the area of the triangle and solve for the height:

$Area =\frac{bh}2{}\Rightarrow 4\sqrt{3}=\frac{4\times h}{2}\Rightarrow h=2\sqrt{3}\ cm$

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Example Question #122 : Isee Upper Level (Grades 9 12) Mathematics Achievement

The height of the equilateral triangle below is 5 centimeters. Give the perimeter of the triangle.

Possible Answers:

$\frac{4\sqrt{3}}{3}\ cm$

$\frac{8\sqrt{3}}{3}\ cm$

$\frac{8}{3}\ cm$

$4\ cm$

$8\ cm$

Correct answer:

$\frac{8\sqrt{3}}{3}\ cm$

Explanation:

In an equilateral triangle all of the sides have the same length, so all we need to do is find the length of one side.

We know that, in an equilateral triangle, all three angles measure $60^{\circ}$ .

$sin\angle A=\frac{h}{AB}=\frac{h}{c}$

$sin\angle A=\frac{h}{c}\Rightarrow sin60^{\circ}=\frac{4}{c}\Rightarrow \frac{\sqrt{3}}{2}=\frac{4}{c}\Rightarrow c=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} \ cm$

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

A convex pentagon has four angles that measure $150^{\circ }$ each. What is the measure of the fifth angle?

Possible Answers:

$90^{\circ }$

$60^{\circ }$

This pentagon cannot exist.

$30^{\circ }$

$120^{\circ }$

Correct answer:

This pentagon cannot exist.

Explanation:

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . If we let the unknown angle measure be $x^{\circ }$ , then from this information:

150 + 150 + 150 + 150 + x = 540

600 + x = 540

600 + x -600 = 540-600

x = -60

Since an angle measure cannot be negative, this pentagon cannot exist.

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Example Question #1 : Pentagons

The measures of the angles of a pentagon are $x^{\circ }, x^{\circ }, x^{\circ }, y^{\circ }, y^{\circ }$ . If y = 100 , what is ?

Possible Answers:

This pentagon cannot exist.

$x = 90^{\circ}$

$x = 113 \frac{1}{3}^{\circ}$

$x = 170^{\circ}$

$x = 53 \frac{1}{3}^{\circ}$

Correct answer:

$x = 113 \frac{1}{3}^{\circ}$

Explanation:

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know that:

x + x + x + y + y = 540

3x + 2y = 540

If y = 100 , then

$3x + 2 \cdot 100 = 540$

3x + 200 = 540

3x + 200 - 200 = 540 - 200

3x = 340

$x = 340 \div 3$

$x = 113 \frac{1}{3}^{\circ}$

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Example Question #2 : Pentagons

The measures of the angles of a pentagon are $x ^{\circ }, x ^{\circ }, x ^{\circ }, y ^{\circ }, y ^{\circ }$ . If $x = 100 ^{\circ }$ , what is ?

Possible Answers:

$y = 80^{\circ}$

$y = 50^{\circ}$

$y = 120^{\circ}$

This pentagon cannot exist.

$y = 30^{\circ}$

Correct answer:

$y = 120^{\circ}$

Explanation:

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know:

x + x + x + y + y = 540

3x + 2y = 540

If $x = 100 ^{\circ }$ , then the above becomes:

$3 \cdot 100 + 2y = 540$

300 + 2y = 540

300 + 2y - 300 = 540 - 300

2y = 240

$2y \div 2 = 240 \div 2$

$y = 120^{\circ}$

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Example Question #3 : Pentagons

Solve for :

Question_11

Possible Answers:

x=87

x=70

x=110

x=90

Correct answer:

x=110

Explanation:

The sum of the interior angles of a pentagon can be determined by the following equation, where is the number of sides:

sum=180(n-2)=180(5-2)=180(3)=540

Therefore:

120+87+90+133+x=540

430+x=540

430+x-430=540-430

x=110

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Example Question #122 : Geometry

Find the perimeter of a pentagon with a side of length 12in.

Possible Answers:

$96\text{in}$

$72\text{in}$

$60\text{in}$

$48\text{in}$

$58\text{in}$

Correct answer:

$60\text{in}$

Explanation:

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon.

Now, we know the length of one side of the pentagon is 12in.

Knowing this, we can substitute into the formula. We get

$P = 5 \cdot 12\text{in}$

$P = 60\text{in}$

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Example Question #122 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Find the perimeter of a pentagon with a side having a length of 21in.

Possible Answers:

$84\text{in}$

$105\text{in}$

$168\text{in}$

$147\text{in}$

$126\text{in}$

Correct answer:

$105\text{in}$

Explanation:

A pentagon has 5 sides. To find the perimeter, we will use the following formula:

P = 5a

where a is the length of a side of the pentagon.

Now, we know the length of a side of the pentagon is 21in.

Knowing this, we will substitute into the formula. We get

$P = 5 \cdot 21\text{in}$

$P = 105\text{in}$

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Example Question #3 : How To Find The Perimeter Of A Pentagon

Find the perimeter of a pentagon with a side having a length of 15cm.

Possible Answers:

$105\text{cm}$

$90\text{cm}$

$60\text{cm}$

$75\text{cm}$

Correct answer:

$75\text{cm}$

Explanation:

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon. Because a pentagon has 5 equal sides, we can use any side in the formula.

So, we know the length of one side of the pentagon is 15cm. So, we can substitute. We get

$P = 5 \cdot 15\text{cm}$

$P = 75\text{cm}$

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Example Question #127 : Isee Upper Level (Grades 9 12) Mathematics Achievement

A pentagon has a side of length 11in. Find the perimeter.

Possible Answers:

$66\text{in}$

$77\text{in}$

$55\text{in}$

$88\text{in}$

$44\text{in}$

Correct answer:

$55\text{in}$

Explanation:

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon.

Now, we know the length of one side of the pentagon is 11in.

Knowing this, we can substitute into the formula. We get

$P = 5 \cdot 11\text{in}$

$P = 55\text{in}$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Plane Geometry

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Find The Height Of An Eqilateral Triangle

Example Question #122 : Isee Upper Level (Grades 9 12) Mathematics Achievement

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

Example Question #1 : Pentagons

Example Question #2 : Pentagons

Example Question #3 : Pentagons

Example Question #122 : Geometry

Example Question #122 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #3 : How To Find The Perimeter Of A Pentagon

Example Question #127 : Isee Upper Level (Grades 9 12) Mathematics Achievement

All ISEE Upper Level Math Resources

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