ISEE Upper Level Quantitative : Geometry

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

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Example Questions

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

Obtuse

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

(a) \(\displaystyle x+ y\)

(b) \(\displaystyle z\)

Possible Answers:

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

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

Extend \(\displaystyle \overline{AB}\) 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,

\(\displaystyle m \angle CAD = m \angle B + m \angle C\),

and

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

However, \(\displaystyle m \angle CAD > z ^{\circ }\), so, by substitution,

\(\displaystyle x+y>z\)

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

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

(a) \(\displaystyle m \angle B\)

(b) \(\displaystyle 60^{\circ }\)

Possible Answers:

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

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

Triangles

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

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

\(\displaystyle m \angle B + m \angle C < 120^{\circ }\)

Substituting and solving:

\(\displaystyle m \angle B + m \angle B < 120^{\circ }\)

\(\displaystyle 2 m \angle B < 120^{\circ }\)

\(\displaystyle 2 m \angle B \div 2 < 120^{\circ } \div 2\)

\(\displaystyle m \angle B < 60^{\circ }\).

 

Example Question #11 : Acute / Obtuse Triangles

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

Which is the greater quantity? 

(a) \(\displaystyle BC\)

(b) \(\displaystyle EF\)

Possible Answers:

(b) is greater.

(a) 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:

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

Ssa

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

Example Question #11 : Geometry

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

Which is the greater quantity? 

(a) \(\displaystyle m \angle A\)

(b) \(\displaystyle m \angle D\)

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:

\(\displaystyle AC = DF, AB = DE, BC = EF\), 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,

\(\displaystyle \Delta ABC \cong \Delta DEF\).

Corresponding angles of congruent sides are congruent, so \(\displaystyle m \angle A = m \angle D\).

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

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

Possible Answers:

\(\displaystyle 5,000 \textrm{ m, } 4,000 \textrm{ m, }8,000 \textrm{ m}\)

\(\displaystyle 6 \textrm{ in, } 8 \textrm{ in, }10 \textrm{ in}\)

\(\displaystyle 0.7 \textrm{ in, } 0.9 \textrm{ in, }1.4 \textrm{ in}\)

\(\displaystyle 8 \textrm{ cm, } 9 \textrm{ cm, }10 \textrm{ cm}\)

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.

Example Question #2 : How To Find The Length Of The Side Of A Triangle

Given \(\displaystyle \Delta ABC\) with right angle \(\displaystyle \angle B\)\(\displaystyle m \angle C = 50 ^{\circ }\)

Which is the greater quantity?

(a) \(\displaystyle AB\)

(b) \(\displaystyle BC\)

Possible Answers:

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

\(\displaystyle m \angle B = 90^{\circ}\)

\(\displaystyle m \angle C = 50 ^{\circ }\)

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

\(\displaystyle m \angle A + m \angle B + m \angle C = 180\)

\(\displaystyle m \angle A +90 + 50 = 180\)

\(\displaystyle m \angle A +140= 180\)

\(\displaystyle m \angle A +140-140= 180-140\)

\(\displaystyle m \angle A = 40^{\circ }\)

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

Example Question #17 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Given \(\displaystyle \Delta ABC\) with obtuse angle \(\displaystyle \angle A\), which is the greater quantity?

(a) \(\displaystyle AB\)

(b) \(\displaystyle BC\)

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

Example Question #18 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

(a) \(\displaystyle AC\)

(b)

Possible Answers:

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

Since \(\displaystyle \angle B\) is the obtuse angle of \(\displaystyle \Delta ABC\)

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

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

\(\displaystyle AC > \sqrt{625}\)

\(\displaystyle AC >25\),

so (a) is the greater quantity.

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

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

(a) \(\displaystyle AC\)

(b) \(\displaystyle 17\)

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

Use the Triangle Inequality:

\(\displaystyle AC < AB + BC\)

\(\displaystyle AC < 5 + 12\)

\(\displaystyle AC < 17\)

This makes (b) the greater quantity.

Example Question #20 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

(a) \(\displaystyle AC\)

(b) \(\displaystyle 13\)

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

By the Converse of the Pythagorean Theorem, 

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

if and only if \(\displaystyle \angle B\) is a right angle. 

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

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

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