Isosceles Triangles

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ISEE Upper Level Quantitative Reasoning › Isosceles Triangles

Questions 1 - 9

Two sides of an isosceles triangle have lengths 3 feet and 4 feet. Which of the following could be the length of the third side?

$36 \textrm{ in}$

$40 \textrm{ in}$

$42 \textrm{ in}$

$32 \textrm{ in}$

$54 \textrm{ in}$

Explanation

An isosceles triangle, by definition, has two sides of equal length. Having the third side measure either 3 feet or 4 feet would make the triangle meet this criterion.

3 feet is equal to $3 \times 12 = 36$ inches, and 4 feet is equal to $4 \times 12 = 48$ inches. We choose 36 inches, since that, but not 48 inches, is a choice.

Two sides of an isosceles triangle have lengths 3 feet and 4 feet. Which of the following could be the length of the third side?

$36 \textrm{ in}$

$40 \textrm{ in}$

$42 \textrm{ in}$

$32 \textrm{ in}$

$54 \textrm{ in}$

Explanation

An isosceles triangle, by definition, has two sides of equal length. Having the third side measure either 3 feet or 4 feet would make the triangle meet this criterion.

3 feet is equal to $3 \times 12 = 36$ inches, and 4 feet is equal to $4 \times 12 = 48$ inches. We choose 36 inches, since that, but not 48 inches, is a choice.

Two sides of an isosceles triangle have lengths 3 feet and 4 feet. Which of the following could be the length of the third side?

$36 \textrm{ in}$

$40 \textrm{ in}$

$42 \textrm{ in}$

$32 \textrm{ in}$

$54 \textrm{ in}$

Explanation

An isosceles triangle, by definition, has two sides of equal length. Having the third side measure either 3 feet or 4 feet would make the triangle meet this criterion.

3 feet is equal to $3 \times 12 = 36$ inches, and 4 feet is equal to $4 \times 12 = 48$ inches. We choose 36 inches, since that, but not 48 inches, is a choice.

One of the base angles of an isosceles triangle is $42^{\circ}$ . Give the measure of the vertex angle.

$96 ^{\circ}$

$98 ^{\circ}$

$100^{\circ}$

$90^{\circ}$

$86 ^{\circ}$

Explanation

The base angles of an isosceles triangle are always equal. Therefore both base angles are $42^{\circ}$ .

Let the measure of the third angle. Since the sum of the angles of a triangle is $180^{\circ}$ , we can solve accordingly:

$42+42+x=180\Rightarrow x=180-84\Rightarrow x=96^{\circ}$

One of the base angles of an isosceles triangle is $42^{\circ}$ . Give the measure of the vertex angle.

$96 ^{\circ}$

$98 ^{\circ}$

$100^{\circ}$

$90^{\circ}$

$86 ^{\circ}$

Explanation

The base angles of an isosceles triangle are always equal. Therefore both base angles are $42^{\circ}$ .

Let the measure of the third angle. Since the sum of the angles of a triangle is $180^{\circ}$ , we can solve accordingly:

$42+42+x=180\Rightarrow x=180-84\Rightarrow x=96^{\circ}$

One of the base angles of an isosceles triangle is $42^{\circ}$ . Give the measure of the vertex angle.

$96 ^{\circ}$

$98 ^{\circ}$

$100^{\circ}$

$90^{\circ}$

$86 ^{\circ}$

Explanation

The base angles of an isosceles triangle are always equal. Therefore both base angles are $42^{\circ}$ .

Let the measure of the third angle. Since the sum of the angles of a triangle is $180^{\circ}$ , we can solve accordingly:

$42+42+x=180\Rightarrow x=180-84\Rightarrow x=96^{\circ}$

The triangles are similar. Solve for .

Question_12

Explanation

Because the triangles are similar, proportions can be used to solve for the length of the side:

$\frac{9}{3}=\frac{6}{x}$

Cross-multiply:

The triangles are similar. Solve for .

Question_12

Explanation

Because the triangles are similar, proportions can be used to solve for the length of the side:

$\frac{9}{3}=\frac{6}{x}$

Cross-multiply:

The triangles are similar. Solve for .

Question_12

Explanation

Because the triangles are similar, proportions can be used to solve for the length of the side:

$\frac{9}{3}=\frac{6}{x}$

Cross-multiply:

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