Math Problem Statement

A triangle has two sides of length 8 and 18. What is the largest possible whole-number length for the third side?

Solution

To find the largest possible whole-number length for the third side of a triangle, we can use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given the two sides are 8 and 18, let's denote the length of the third side as xx. We have three inequalities based on the triangle inequality:

  1. 8+18>x8 + 18 > x (the sum of the two known sides must be greater than the third side)
  2. 8+x>188 + x > 18 (the sum of one known side and the third side must be greater than the other known side)
  3. 18+x>818 + x > 8 (the sum of the other known side and the third side must be greater than the first known side)

Solving each inequality:

  1. From 8+18>x8 + 18 > x, we get: 26>xorx<2626 > x \quad \text{or} \quad x < 26

  2. From 8+x>188 + x > 18, we get: x>188orx>10x > 18 - 8 \quad \text{or} \quad x > 10

  3. From 18+x>818 + x > 8, we get: x>818orx>10x > 8 - 18 \quad \text{or} \quad x > -10 But this condition is automatically satisfied since x>10x > 10.

Combining the conditions:

The third side must satisfy: 10<x<2610 < x < 26

Therefore, the largest whole-number value for xx is 25.

Let me know if you'd like further clarification or help with another problem!

Follow-up questions:

  1. How would the smallest possible third side be calculated using the triangle inequality?
  2. Can the triangle inequality theorem be applied to non-triangular shapes?
  3. How can we calculate the area of a triangle given two sides and the included angle?
  4. What happens to the shape of a triangle when the third side is very small or very large?
  5. How does the Pythagorean theorem relate to right triangles in the context of side lengths?

Tip:

Always check if a triangle inequality holds before attempting to calculate angles, area, or other properties.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Triangle Inequality
Geometry

Formulas

Triangle Inequality: a + b > c

Theorems

Triangle Inequality Theorem

Suitable Grade Level

Grades 6-8