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ISEE Upper Level Math : Geometry

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

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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 If Two Acute / Obtuse Triangles Are Similar

Similar

NOTE: Figures NOT drawn to scale.

Refer to the above two triangles.

$\Delta ABC \sim \Delta DEF ,AB = 25, DE = 20, AC = 20$

What is ?

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

Corresponding sides of similar triangle are proportional, so if

$\Delta ABC \sim \Delta DEF$ , then

$\frac{DF}{AC} = \frac{DE}{AB}$

Substitute the known sidelengths, then solve for :

$\frac{DF}{20} = \frac{20}{25}$

$\frac{DF}{20}\cdot 20= \frac{20}{25}\cdot 20$

$DF = \frac{400}{25}$

DF =16

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Example Question #2 : Acute / Obtuse Triangles

$\Delta ABC \sim \Delta DEF$

AB = 15, BC = 21, AC = 12, DF = 36

What is the perimeter of $\Delta DEF$ ?

Possible Answers:

It cannot be determined from the information given.

150

144

136

128

Correct answer:

144

Explanation:

By definition, since, $\Delta ABC \sim \Delta DEF$ , side lengths are in proportion.

So,

$\frac{DE}{AB}= \frac{DF}{AC}$

$\frac{DE}{15}= \frac{36}{12} = 3$

$\frac{DE}{15} \cdot 15 = 3\cdot 15$

DE = 45

$\frac{EF}{BC}= \frac{DF}{AC}$

$\frac{EF}{21}= \frac{36}{12} = 3$

$\frac{EF}{21} \cdot 21 = 3\cdot 21$

EF = 63

The perimeter of $\Delta DEF$ is

DE + EF + DF = 45 + 63 + 36 = 144 .

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Example Question #2 : Acute / Obtuse Triangles

$\Delta ABC \sim \Delta DEF$

AB = 15, BC = 19, DE = 45, DF = 66

What is ?

Possible Answers:

AC = 22

It is impossible to tell from the information given.

AC = 198

AC = 57

AC = 15

Correct answer:

AC = 22

Explanation:

By definition, since $\Delta ABC \sim \Delta DEF$ , all side lengths are in proportion.

$\frac{AC}{DF} = \frac{AB}{DE}$

$\frac{AC}{66} = \frac{15}{45}$

$\frac{AC}{66} = \frac{1}{3}$

$AC = \frac{1}{3} \cdot 66 = 22$

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

Which of the following is true about a triangle with two angles that measure $120^{\circ }$ and $90^{\circ }$ ?

Possible Answers:

This triangle is scalene and right.

This triangle is scalene and obtuse.

This triangle is isosceles and obtuse.

This triangle cannot exist.

This triangle is isosceles and right.

Correct answer:

This triangle cannot exist.

Explanation:

A triangle must have at least two acute angles; however, a triangle with angles that measure $120^{\circ }$ and $90^{\circ }$ could have at most one acute angle, an impossible situation. Therefore, this triangle is nonexistent.

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Example Question #4 : Acute / Obtuse Triangles

Which of the following is true about a triangle with two angles that measure $91^{\circ }$ each?

Possible Answers:

The triangle is acute and isosceles.

The triangle is acute and scalene.

The triangle is obtuse and isosceles.

The triangle is obtuse and scalene.

The triangle cannot exist.

Correct answer:

The triangle cannot exist.

Explanation:

A triangle must have at least two acute angles; however, a triangle with angles that measure $91 ^{\circ }$ would have two obtuse angles and at most one acute angle. This is not possible, so this triangle cannot exist.

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Example Question #5 : Acute / Obtuse Triangles

One angle of an isosceles triangle has measure $110^{\circ }$ . What are the measures of the other two angles?

Possible Answers:

$70^{\circ }, 70^{\circ }$

$110 ^{\circ }, 40 ^{\circ }$

$35^{\circ },35^{\circ }$

$45^{\circ }, 45^{\circ }$

Not enough information is given to answer this question.

Correct answer:

$35^{\circ },35^{\circ }$

Explanation:

An isosceles triangle not only has two sides of equal measure, it has two angles of equal measure. This means one of two things, which we examine separately:

Case 1: It has another $110^{\circ }$ angle. This is impossible, since a triangle cannot have two obtuse angles.

Case 2: Its other two angles are the ones that are of equal measure. If we let be their common measure, then, since the sum of the measures of a triangle is $180^{\circ }$ ,

x + x + 110 = 180

2x + 110 = 180

2x + 110 -110= 180 -110

2x= 70

$2x \div 2= 70 \div 2$

x = 35

Both angles measure $35^{\circ}$

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Example Question #2 : Acute / Obtuse Triangles

The angles of a triangle measure $x^{\circ }, x^{\circ },\left ( x -42 \right ) ^{\circ }$ . Evaluate .

Possible Answers:

x = 72

x = 68

x = 74

x = 70

Correct answer:

x = 74

Explanation:

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

x + x+ ( x -42 ) = 180

3x -42 = 180

3x -42+42 = 180+42

3x = 222

$3x \div 3 = 222 \div 3$

x = 74

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Example Question #3 : Acute / Obtuse Triangles

The acute angles of a right triangle measure $\left ( 2x - 8 \right )^{\circ }$ and $\left ( 3x - 4 \right )^{\circ }$ .

Evaluate .

Possible Answers:

x= 20.4

x= 38.4

x = 15.6

x = 33.6

Correct answer:

x= 20.4

Explanation:

The degree measures of the acute angles of a right triangle total 90, so we solve for in the following equation:

$\left ( 2x - 8 \right ) + \left ( 3x - 4 \right ) = 90$

5x - 12 = 90

5x - 12+ 12 = 90 + 12

5x = 102

$5x \div 5 = 102 \div 5$

x= 20.4

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Example Question #4 : Acute / Obtuse Triangles

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. $\overline{AB } \cong \overline{AC}$ ; $m \widehat{AC} = 140 ^{\circ }$ .

What is the measure of $\angle A$ ?

Possible Answers:

$40 ^{\circ }$

$60 ^{\circ }$

$50 ^{\circ }$

$70 ^{\circ }$

Correct answer:

$40 ^{\circ }$

Explanation:

Congruent chords of a circle have congruent minor arcs, so since $\overline{AB } \cong \overline{AC}$ , $\widehat{AB } \cong \widehat{AC}$ , and their common measure is $m\widehat{AB } =m \widehat{AC} = 140^{\circ}$ .

Since there are $360^{\circ}$ in a circle,

$m\widehat{BC } + m\widehat{AB } + m \widehat{AC} = 360^{\circ}$

$m\widehat{BC } + 140^{\circ} + 140^{\circ}= 360^{\circ}$

$m\widehat{BC } + 280^{\circ}= 360^{\circ}$

$m\widehat{BC } = 80^{\circ}$

The inscribed angle $\angle A$ intercepts this arc and therefore has one-half its degree measure, which is $40 ^{\circ }$

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Example Question #5 : Acute / Obtuse Triangles

Solve for :
Question11

Possible Answers:

x=44

x=46

x=134

x=226

Correct answer:

x=46

Explanation:

The sum of the internal angles of a triangle is equal to 180^o . Therefore:

180=45+89+x

180=134+x

180-134=134-134+x

46=x

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : 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 If Two Acute / Obtuse Triangles Are Similar

Example Question #2 : Acute / Obtuse Triangles

Example Question #2 : Acute / Obtuse Triangles

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

Example Question #4 : Acute / Obtuse Triangles

Example Question #5 : Acute / Obtuse Triangles

Example Question #2 : Acute / Obtuse Triangles

Example Question #3 : Acute / Obtuse Triangles

Example Question #4 : Acute / Obtuse Triangles

Example Question #5 : Acute / Obtuse Triangles

All ISEE Upper Level Math Resources

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