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ISEE Upper Level Math : Acute / Obtuse Triangles

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Example Question #1 : Acute / Obtuse Triangles

Similar

NOTE: Figures NOT drawn to scale.

Refer to the above two triangles.

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

What is $\displaystyle DF$?

Possible Answers:

$\displaystyle 25$

Insufficient information is given to answer the question.

$\displaystyle 18$

$\displaystyle 15$

$\displaystyle 16$

Correct answer:

$\displaystyle 16$

Explanation:

Corresponding sides of similar triangle are proportional, so if

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

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

Substitute the known sidelengths, then solve for $\displaystyle DF$:

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

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

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

DF =16 $\displaystyle DF =16$

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Example Question #1 : Acute / Obtuse Triangles

$\Delta ABC \sim \Delta DEF$ $\displaystyle \Delta ABC \sim \Delta DEF$

$\displaystyle AB = 15, BC = 21, AC = 12, DF = 36$

What is the perimeter of $\Delta DEF$ $\displaystyle \Delta DEF$ ?

Possible Answers:

128 $\displaystyle 128$

136 $\displaystyle 136$

150 $\displaystyle 150$

144 $\displaystyle 144$

It cannot be determined from the information given.

Correct answer:

144 $\displaystyle 144$

Explanation:

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

So,

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

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

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

DE = 45 $\displaystyle DE = 45$

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

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

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

EF = 63 $\displaystyle EF = 63$

The perimeter of $\Delta DEF$ $\displaystyle \Delta DEF$ is

$\displaystyle DE + EF + DF = 45 + 63 + 36 = 144$.

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Example Question #2 : Acute / Obtuse Triangles

$\Delta ABC \sim \Delta DEF$ $\displaystyle \Delta ABC \sim \Delta DEF$

$\displaystyle AB = 15, BC = 19, DE = 45, DF = 66$

What is $\displaystyle AC$ ?

Possible Answers:

AC = 57 $\displaystyle AC = 57$

AC = 22 $\displaystyle AC = 22$

AC = 15 $\displaystyle AC = 15$

It is impossible to tell from the information given.

$\displaystyle AC = 198$

Correct answer:

AC = 22 $\displaystyle AC = 22$

Explanation:

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

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

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

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

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

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Example Question #2 : Acute / Obtuse Triangles

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

Possible Answers:

This triangle is isosceles and obtuse.

This triangle is scalene and right.

This triangle is isosceles and right.

This triangle cannot exist.

This triangle is scalene and obtuse.

Correct answer:

This triangle cannot exist.

Explanation:

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

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Example Question #3 : Acute / Obtuse Triangles

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

Possible Answers:

The triangle cannot exist.

The triangle is obtuse and scalene.

The triangle is obtuse and isosceles.

The triangle is acute and scalene.

The triangle is acute and isosceles.

Correct answer:

The triangle cannot exist.

Explanation:

A triangle must have at least two acute angles; however, a triangle with angles that measure $91 ^{\circ }$ $\displaystyle 91 ^{\circ }$ 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 #1 : Acute / Obtuse Triangles

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

Possible Answers:

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

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

Not enough information is given to answer this question.

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

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

Correct answer:

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

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 $110^{\circ }$ $\displaystyle 110^{\circ }$ 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 $\displaystyle x$ be their common measure, then, since the sum of the measures of a triangle is $180^{\circ }$ $\displaystyle 180^{\circ }$,

$\displaystyle x + x + 110 = 180$

$\displaystyle 2x + 110 = 180$

$\displaystyle 2x + 110 -110= 180 -110$

2x= 70 $\displaystyle 2x= 70$

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

x = 35 $\displaystyle x = 35$

Both angles measure $35^{\circ}$ $\displaystyle 35^{\circ}$

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Example Question #2 : Acute / Obtuse Triangles

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

Possible Answers:

x = 72 $\displaystyle x = 72$

x = 68 $\displaystyle x = 68$

x = 74 $\displaystyle x = 74$

x = 70 $\displaystyle x = 70$

Correct answer:

x = 74 $\displaystyle x = 74$

Explanation:

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

$\displaystyle x + x+ ( x -42 ) = 180$

$\displaystyle 3x -42 = 180$

$\displaystyle 3x -42+42 = 180+42$

$\displaystyle 3x = 222$

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

x = 74 $\displaystyle x = 74$

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Example Question #3 : Acute / Obtuse Triangles

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

Evaluate $\displaystyle x$.

Possible Answers:

x= 20.4 $\displaystyle x= 20.4$

x= 38.4 $\displaystyle x= 38.4$

$\displaystyle x = 15.6$

$\displaystyle x = 33.6$

Correct answer:

x= 20.4 $\displaystyle x= 20.4$

Explanation:

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

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

$\displaystyle 5x - 12 = 90$

$\displaystyle 5x - 12+ 12 = 90 + 12$

$\displaystyle 5x = 102$

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

x= 20.4 $\displaystyle x= 20.4$

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Example Question #4 : Acute / Obtuse Triangles

Chords

Note: Figure NOT drawn to scale

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

What is the measure of $\angle A$ $\displaystyle \angle A$ ?

Possible Answers:

$40 ^{\circ }$ $\displaystyle 40 ^{\circ }$

$60 ^{\circ }$ $\displaystyle 60 ^{\circ }$

$50 ^{\circ }$ $\displaystyle 50 ^{\circ }$

$70 ^{\circ }$ $\displaystyle 70 ^{\circ }$

Correct answer:

$40 ^{\circ }$ $\displaystyle 40 ^{\circ }$

Explanation:

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

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

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

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

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

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

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

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Example Question #4 : Acute / Obtuse Triangles

Solve for $\displaystyle x$:
Question11

Possible Answers:

x=44 $\displaystyle x=44$

x=134 $\displaystyle x=134$

x=226 $\displaystyle x=226$

x=46 $\displaystyle x=46$

Correct answer:

x=46 $\displaystyle x=46$

Explanation:

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

$\displaystyle 180=45+89+x$

$\displaystyle 180=134+x$

$\displaystyle 180-134=134-134+x$

46=x $\displaystyle 46=x$

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ISEE Upper Level Math : Acute / Obtuse Triangles

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Acute / Obtuse Triangles

Example Question #1 : Acute / Obtuse Triangles

Example Question #2 : Acute / Obtuse Triangles

Example Question #2 : Acute / Obtuse Triangles

Example Question #3 : Acute / Obtuse Triangles

Example Question #1 : Acute / Obtuse Triangles

Example Question #2 : Acute / Obtuse Triangles

Example Question #3 : Acute / Obtuse Triangles

Example Question #4 : Acute / Obtuse Triangles

Example Question #4 : Acute / Obtuse Triangles

All ISEE Upper Level Math Resources

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