ISEE Upper Level Quantitative : How to find the length of the side of a triangle

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?

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

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  with right angle

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

(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 #3 : How To Find The Length Of The Side Of A Triangle

Given  with obtuse angle , which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

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

has obtuse angle . Which is the greater quantity?

(a)

(b)

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(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)

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Explanation:

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)

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

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.

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

is acute; . Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

(b) is greater.

Explanation:

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)

(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

(a) is the greater quantity

Explanation:

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,

.