ISEE Upper Level Quantitative : Triangles

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) 

(b) 

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

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

Correct answer:

(a) is the greater quantity

Explanation:

Extend  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,

,

and

 

However, , so, by substitution,

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

Given: . Which is the greater quantity?

(a) 

(b) 

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 , an equilateral triangle with sides of length 10:

Triangles

As an angle of an equilateral triangle,  has measure . Applying the Side-Side-Side Inequality Theorem, since , and , it follows that , so .

Also, since , by the Isosceles Triangle Theorem, . Since , and the sum of the measures of the angles of a triangle is , it follows that

Substituting and solving:

.

 

Example Question #1 : How To Find If Triangles Are Congruent

Given  and  with 

Which is the greater quantity? 

(a) 

(b) 

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(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  and . The two question marks need to be replaced by  and . No matter how you place these two points, . However, with one replacement, ; with the other replacement, . Therefore, the information given is insufficient to answer the question.

Example Question #11 : Geometry

Consider  and  with .

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) and (b) are equal.

Explanation:

, 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,

.

Corresponding angles of congruent sides are congruent, so .

Example Question #12 : 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:

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

Given  with right angle 

Which is the greater quantity?

(a) 

(b) 

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:

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

, so the side opposite , which is , is longer than the side opposite , which is . This makes (a) the greater quantity.

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

Given  with obtuse angle , 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  and  from the angle measures, it is necessary to know which of their opposite angles -  and , respectively - is the greater angle. Since  is the obtuse angle, it has the greater measure, and  is the longer side. This makes (b) greater.

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

 has obtuse angle . Which is the greater quantity?

(a) 

(b)

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

Since  is the obtuse angle of 

.

,

,

so (a) is the greater quantity.

Example Question #5 : How To Find The Length Of The Side Of A Triangle

Given  with . 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:

Use the Triangle Inequality:

This makes (b) the greater quantity.

Example Question #13 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Given  with . Which is the greater quantity?

(a) 

(b)

Possible Answers:

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

It is impossible to tell from the information given.

Explanation:

By the Converse of the Pythagorean Theorem, 

if and only if  is a right angle. 

However, if  is acute, then ;  if  is obtuse, then .

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

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