Triangle Inequality Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
In △ABC\triangle ABC, AB=6AB = 6 and BC=10BC = 10. The perimeter is 24. Is this a valid triangle? Find ACAC and verify.

Solution

  1. 1
    Step 1: Find ACAC: Perimeter =AB+BC+AC=6+10+AC=24= AB + BC + AC = 6 + 10 + AC = 24, so AC=24−16=8AC = 24 - 16 = 8.
  2. 2
    Step 2: Check the triangle inequality with sides 6, 8, 10: 6+8=14>106 + 8 = 14 > 10 ✓; 6+10=16>86 + 10 = 16 > 8 ✓; 8+10=18>68 + 10 = 18 > 6 ✓.
  3. 3
    Step 3: Also notice 62+82=36+64=100=1026^2 + 8^2 = 36 + 64 = 100 = 10^2, so this is a right triangle!

Answer

AC=8AC = 8; the triangle is valid (and is a 6-8-10 right triangle).
First use the perimeter to find the missing side, then verify the triangle inequality. The sides 6, 8, 10 form a valid triangle. As a bonus, recognizing that 62+82=1026^2 + 8^2 = 10^2 reveals this is a Pythagorean triple (a scaled 3-4-5 right triangle).

About Triangle Inequality

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

Learn more about Triangle Inequality →

More Triangle Inequality Examples

Example 1 easy

Can sides of length 4, 7, and 10 form a triangle? Check all three triangle inequality conditions.

Example 2 medium

Two sides of a triangle have lengths 8 and 13. Find the range of possible lengths for the third side

Example 3 easy

Which of these sets of side lengths cannot form a triangle? (a) 3, 4, 5. (b) 1, 2, 3. (c) 5, 8, 12.

← All Triangle Inequality Examples Triangle Inequality Concept