### All ISEE Upper Level Quantitative Resources

## Example Questions

### 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:**

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

Given with right angle , .

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

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

### Example Question #3 : 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.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

**Correct answer:**

(b) is greater.

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

has obtuse angle ; . Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

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:**

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

**Correct answer:**

(b) is greater.

Use the Triangle Inequality:

This makes (b) the greater quantity.

### Example Question #6 : 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:**

It is impossible to tell from the information given.

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.

### Example Question #7 : How To Find The Length Of The Side Of A Triangle

is acute; . Which is the greater quantity?

(a)

(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.

Since is an acute triangle, is an acute angle, and

,

(b) is the greater quantity.

### Example Question #8 : How To Find The Length Of The Side Of A Triangle

Given: . . Which is the greater quantity?

(a) 18

(b)

**Possible Answers:**

(a) and (b) are equal

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

(b) is the greater quantity

(a) is the greater quantity

**Correct answer:**

(a) is the greater quantity

Suppose there exists a second triangle such that and . Whether , the angle opposite the longest side, 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:

, making obtuse, so .

We know that

and

.

Between and , 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, is the longer. Therefore,

.