2005 AMC 10B Problem 10

Problem 10 of 25EasierGeometry

In

\triangle ABC,

we have

AC = BC = 7

and

AB = 2.

Suppose that

D

is a point on line

AB

such that

B

lies between

A

and

D

and

CD = 8.

What is

BD?

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Solution

Let

H

be the foot of the altitude from

C

to line

AB.

Since

AC = BC,

H

is the midpoint of

AB,

so

BH = 1

and

CH^2 = 7^2 - 1^2 = 48.

Applying the Pythagorean theorem in

\triangle CHD,

where

HD = BH + BD = 1 + BD,

gives

8^2 = 48 + (1 + BD)^2,

so

(1 + BD)^2 = 16.

Then

1 + BD = 4,

so

BD = 3.

Thus, A is the correct answer.

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Concepts: isosceles triangle · altitude · Pythagorean Theorem

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.