Question

The lengths of the sides of a triangle are integers, and its area is also an integer. One side is $21$ and the perimeter is $48.$ The shortest side is:

$8$

$10$

$12$

$14$

$16$

Answer: B

Concepts:Heron’s Formula casework triangle inequality

Difficulty rating: 2100

Small Hint:

Write the other sides as $x$ and $27-x$ , with semiperimeter $24$

Big Hint:

Heron’s formula makes the squared area $72(24-x)(x-3)$

Solution:

Let the other sides be $x$ and $27-x,$ with $x\leq27-x.$ The semiperimeter is $24,$ so Heron’s formula gives $\begin{aligned} K^2 &=24\cdot3(24-x)(x-3)\\ &=72(24-x)(x-3). \end{aligned}$ The triangle inequalities give $4\leq x\leq13.$ Checking these integers, the expression is not a square for $x=4,\ldots,9,$ while at $x=10$ it is $72\cdot14\cdot7=7056=84^2.$ Thus the shortest side is $10.$

Therefore, the correct answer is B.

1963 AMC 12 Problem 35

Answer: B

Difficulty rating: 2100

Small Hint:

Big Hint:

Solution:

Problem 35 in Other Years