1988 AMC 8 Problem 9

Question

An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?

1

2

3

4

5

Answer: D

Concepts:isosceles triangle distance formula

Difficulty rating: 960

Small hint:

Find each triangle’s side lengths from the grid: a slanted side spanning

a

across and

b

up has length

\sqrt{a^2 + b^2}.

A triangle is isosceles when two of its sides are equal.

Big hint:

Work out all three side lengths for each of the five triangles and look for a matching pair; only one triangle ends up with three different side lengths.

Solution:

Read each triangle’s side lengths from the grid. A horizontal or vertical side is counted directly, and a slanted side spanning

a

units across and

b

units up has length

\sqrt{a^2 + b^2}.

Top-left triangle: its two slanted sides each span

1

across and

2

up, so each has length

\sqrt{5},

with a base of length

2.

Two equal sides, so it is isosceles.

Top-middle triangle: it has a vertical side of length

2

and a horizontal side of length

2

(its third side is the slant

2\sqrt{2}

). Two equal sides, so it is isosceles.

Top-right triangle: its sides are

2,

\sqrt{5},

and

\sqrt{13},

which are all different, so it is not isosceles.

Bottom-left (wide, flat) triangle: its two slanted sides each span

3

across and

1

up, so each has length

\sqrt{10}.

Two equal sides, so it is isosceles.

Bottom-right triangle: two of its sides each span

2

and

1

(giving length

\sqrt{5}

), while the third is

\sqrt{10}.

Two equal sides, so it is isosceles.

Four of the five triangles are isosceles.

Thus, the correct answer is D .

1988 AMC 8 Problem 9

Answer: D

Difficulty rating: 960

Small hint:

Big hint:

Solution:

Problem 9 in other years