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ISEE Upper Level Quantitative : Geometry

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

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All ISEE Upper Level Quantitative Resources

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Example Questions

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Example Question #11 : Plane Geometry

Obtuse

Refer to the above figure. Which is the greater quantity?

(a) x+ y

(b)

Possible Answers:

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

Extend $\overline{AB}$ as seen in the figure below:

Obtuse

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles; specifically,

$m \angle CAD = m \angle B + m \angle C$ ,

and

$m \angle CAD =( x+ y) ^{\circ }$

However, $m \angle CAD > z ^{\circ }$ , so, by substitution,

x+y>z

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Example Question #12 : Acute / Obtuse Triangles

Given: $\bigtriangleup ABC$ . AB= AC = 10, BC = 12 . Which is the greater quantity?

(a) $m \angle B$

(b) $60^{\circ }$

Possible Answers:

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

Below is the referenced triangle along with $\bigtriangleup DEF$ , an equilateral triangle with sides of length 10:

Triangles

As an angle of an equilateral triangle, $\angle D$ has measure $60 ^{\circ }$ . Applying the Side-Side-Side Inequality Theorem, since AB = DE , AC = DF , and BC > EF , it follows that $m \angle A > m \angle D$ , so $m \angle A > 60^{\circ }$ .

Also, since AB = BC , by the Isosceles Triangle Theorem, $m \angle B = m \angle C$ . Since $m \angle A > 60^{\circ }$ , and the sum of the measures of the angles of a triangle is $180 ^{\circ }$ , it follows that

$m \angle B + m \angle C < 120^{\circ }$

Substituting and solving:

$m \angle B + m \angle B < 120^{\circ }$

$2 m \angle B < 120^{\circ }$

$2 m \angle B \div 2 < 120^{\circ } \div 2$

$m \angle B < 60^{\circ }$ .

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Example Question #13 : Acute / Obtuse Triangles

Given $\Delta ABC$ and $\Delta DEF$ with $AC = DF, AB = DE, m \angle B = m \angle E$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Examine the diagram below, in which two triangles matching the given descriptions have been superimposed.

Ssa

Note that AB = DE and $m \angle B = m \angle E$ . The two question marks need to be replaced by and . No matter how you place these two points, AC = DF . However, with one replacement, BC > EF ; with the other replacement, BC < EF . Therefore, the information given is insufficient to answer the question.

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Example Question #14 : Acute / Obtuse Triangles

Consider $\Delta ABC$ and $\Delta DEF$ with AC = DF, AB = DE, BC = EF .

Which is the greater quantity?

(a) $m \angle A$

(b) $m \angle D$

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

Correct answer:

(a) and (b) are equal.

Explanation:

AC = DF, AB = DE, BC = EF , so, by the Side-Side-Side Principle, since there are three pairs of congruent corresponding sides between the triangles, we can say they are congruent - that is,

$\Delta ABC \cong \Delta DEF$ .

Corresponding angles of congruent sides are congruent, so $m \angle A = m \angle D$ .

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Example Question #1 : How To Find The Length Of The Side Of A Triangle

Which of the following could be the lengths of the three sides of a scalene triangle?

Possible Answers:

$0.7 \textrm{ in, } 0.9 \textrm{ in, }1.4 \textrm{ in}$

$6 \textrm{ in, } 8 \textrm{ in, }10 \textrm{ in}$

$5,000 \textrm{ m, } 4,000 \textrm{ m, }8,000 \textrm{ m}$

$8 \textrm{ cm, } 9 \textrm{ cm, }10 \textrm{ cm}$

All of the other choices are possible lengths of a scalene triangle

Correct answer:

All of the other choices are possible lengths of a scalene triangle

Explanation:

A scalene triangle, by definition, has sides all of different lengths. Since all of the given choices fit that criterion, the correct choice is that all can be scalene.

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Example Question #16 : Acute / Obtuse Triangles

Given $\Delta ABC$ with right angle $\angle B$ , $m \angle C = 50 ^{\circ }$ .

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

Correct answer:

(a) is greater

Explanation:

$m \angle B = 90^{\circ}$

$m \angle C = 50 ^{\circ }$

The sum of the measures of the angles of a triangle is 180, so

$m \angle A + m \angle B + m \angle C = 180$

$m \angle A +90 + 50 = 180$

$m \angle A +140= 180$

$m \angle A +140-140= 180-140$

$m \angle A = 40^{\circ }$

$m \angle C > m \angle A$ , so the side opposite $\angle C$ , which is $\overline{AB}$ , is longer than the side opposite $\angle A$ , which is $\overline{BC}$ . This makes (a) the greater quantity.

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Example Question #11 : Triangles

Given $\Delta ABC$ with obtuse angle $\angle A$ , which is the greater quantity?

(a)

(b)

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

To compare the lengths of $\overline{AB}$ and $\overline{BC}$ from the angle measures, it is necessary to know which of their opposite angles - $\angleC$ $\angle C$ and $\angle A$ , respectively - is the greater angle. Since $\angle A$ is the obtuse angle, it has the greater measure, and $\overline{BC}$ is the longer side. This makes (b) greater.

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Example Question #1 : How To Find The Length Of The Side Of A Triangle

$\Delta ABC$ has obtuse angle $\angle B$ ; AB = 7, BC = 24 . Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

Since $\angle B$ is the obtuse angle of $\Delta ABC$ ,

$\left ( AC \right )^{2} >\left ( AB \right )^{2} + \left ( BC \right )^{2} = 7^{2} + 24^{2} = 49 + 576 = 625$ .

$\left ( AC \right )^{2} > 625$ ,

$AC > \sqrt{625}$

AC >25 ,

so (a) is the greater quantity.

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Example Question #11 : Plane Geometry

Given $\Delta ABC$ with AB = 5, BC = 12 . Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

Use the Triangle Inequality:

AC < AB + BC

AC < 5 + 12

AC < 17

This makes (b) the greater quantity.

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Example Question #11 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Given $\Delta ABC$ with AB = 5, BC = 12 . Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

It is impossible to tell from the information given.

Explanation:

By the Converse of the Pythagorean Theorem,

$AC = \sqrt{(AB)^{2}+(BC)^{2}}= \sqrt{5^{2}+12^{2}}= \sqrt{25+144}=\sqrt{169} = 13$

if and only if $\angle B$ is a right angle.

However, if $\angle B$ is acute, then AC < 13 ; if $\angle B$ is obtuse, then AC > 13 .

Since we do not know whether $\angle B$ is acute, right, or obtuse, we cannot determine whether (a) or (b) is greater.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Geometry

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #11 : Plane Geometry

Example Question #12 : Acute / Obtuse Triangles

Example Question #13 : Acute / Obtuse Triangles

Example Question #14 : Acute / Obtuse Triangles

Example Question #1 : How To Find The Length Of The Side Of A Triangle

Example Question #16 : Acute / Obtuse Triangles

Example Question #11 : Triangles

Example Question #1 : How To Find The Length Of The Side Of A Triangle

Example Question #11 : Plane Geometry

Example Question #11 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

All ISEE Upper Level Quantitative Resources

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