Question

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 $BD?$

$3$

$2\sqrt{3}$

$4$

$5$

$4\sqrt{2}$

Answer: A

Concepts:isosceles triangle altitude Pythagorean Theorem

Difficulty rating: 1350

Solution:

Let $H$ be the foot of the altitude from $C$ to line $AB.$ Since $\triangle ABC$ is isosceles with $AC = BC,$ $H$ is the midpoint of $AB,$ so $AH = HB = 1.$

Then $CH^2 = 7^2 - 1^2 = 48.$ Applying the Pythagorean Theorem to $\triangle CHD$ with $HD = HB + BD = 1 + BD$ gives $8^2 = 48 + (1 + BD)^2,$ so $(1 + BD)^2 = 16.$

Therefore $1 + BD = 4,$ which means $BD = 3.$

Thus, the correct answer is A.

2005 AMC 12B Problem 6

Answer: A

Difficulty rating: 1350

Solution:

Problem 6 in Other Years