Triangle Inequality (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Triangle Inequality: core idea

Perimeter bounds.

How many integer values of $x$ make $7,10,x$ a triangle?

Hint: Combine side bounds with angle and right-triangle facts. Answer: 13.

1. A triangle has integer sides $4,9,x$ . Find the sum of all possible $x$ .
2. The two shorter sides of a right triangle are $5$ and $12$ . Find the third side.
3. A triangle has sides $8,15,17$ . Is it right? Type $1$ for yes.
4. Find the largest angle's opposite side in a triangle with sides $6,9,11$ .
5. An isosceles triangle has equal sides $10$ and base $12$ . Find its height.
6. How many diagonals does a triangle have?
7. A triangle has angles $45^\circ$ and $65^\circ$ . Find its third angle.
8. A triangle's exterior angle is $120^\circ$ and one remote interior angle is $50^\circ$ . Find the other.
9. Find the area of an isosceles triangle with sides $5,5,6$ .