### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #15 : Acute / Obtuse Triangles

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

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 #16 : Acute / Obtuse Triangles

GivenÂ Â with right angleÂ ,Â .Â

Which is the greater quantity?

(a)Â

(b)Â

**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

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

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

### Example Question #11 : Triangles

GivenÂ Â with obtuseÂ angleÂ , which is the greater quantity?

(a)Â

(b)Â

**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.

To compare the lengths ofÂ Â andÂ Â from the angle measures, it is necessary to know which of their opposite angles -Â Â and , respectively - is the greater angle.Â SinceÂ Â is the obtuse angle, it has the greater measure, and Â is the longer side. This makes (b) greater.

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

Â hasÂ obtuse angleÂ ;Â . Which is the greater quantity?

(a)Â

(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.

SinceÂ Â is theÂ obtuseÂ angle ofÂ ,Â

.

,

,

so (a) is the greater quantity.

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

GivenÂ Â withÂ . Which is the greater quantity?

(a)Â

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

**Correct answer:**

(b) is greater.

Use the Triangle Inequality:

This makes (b) the greater quantity.

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

GivenÂ Â withÂ . Which is the greater quantity?

(a)Â

(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.

By the Converse of the Pythagorean Theorem,Â

if and only ifÂ Â is a right angle.Â

However, ifÂ Â is acute, thenÂ ;Â Â ifÂ Â is obtuse, thenÂ .

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

### Example Question #21 : Triangles

Â is acute;Â . Which is the greater quantity?

(a)Â

(b)Â

**Possible Answers:**

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(b) is greater.

SinceÂ Â is an acute triangle,Â Â is an acute angle, andÂ

,

(b) is the greaterÂ quantity.

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

Given:Â .Â . Which is the greater quantity?

(a) 18

(b)Â

**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

Suppose there exists a second triangleÂ Â such thatÂ Â andÂ . WhetherÂ , the angle opposite the longest side,Â is acute, right, or obtuse can be determined by comparing the sum of the squares of the lengths of the shortest sides to the square of the length of the longest:

, makingÂ Â obtuse, soÂ .

We know that

and

.

Between Â andÂ , we have two sets of congruent sides, with the included angle of the latter of greater measure than that of the former. It follows from the Side-Angle-Side Inequality (or Hinge) Theorem that between the third sides,Â Â is the longer. Therefore,Â

.