Question

The three sides of a triangle have lengths of 8, 12, and m. Which of the following could be the area of the triangle?

1. 28
2. 48
3. 54

A) 1 ONLY
B) 2 ONLY
C) 1 AND 2 ONLY
D) 2 AND 3 ONLY
E) 1, 2, AND 3 ONLY

Answer 1

@ganeshie8 @abishar @hartnn @perl

Answer 2

The largest possible area would be a right triangle:

The smallest would be a very narrow wedge, and can be as close to 0 as you want:


Any area between those two are possible.

Answer 3

Can you find the area of the right triangle? Any area less than or equal to that is acceptable.

Answer 4

Explanation directly from the test prep book, and why I don't understand it:

"Consider the side of length 8 to be the base and attach the side of length 12. Notice that the triangle has the greatest possible height when the two sides form a right angle. Therefore, the greatest possible area of such a triangle is 1/2(12)(8) = 48, and the minimum possible area is 0. Statements 1 and 2 are possible.

Answer 5

While I understand the reasoning and logic, even though it is a right angle, why is that number the maximum?

Answer 6

Because it is :P

I'm sure there's some proof for it somewhere, but for now all you need to remember is that the largest triangle given two side lengths is a right triangle.

Answer 7

\[A = \frac{1}{2}ab\sin\theta\]

Answer 8

Notice the maximum value of \(\sin \theta\) is 1 and it is achieved when \(\theta = 90^{\circ}\)