Math Problem Statement

Given AB = x + 2, AD = x - 2, DC = x + 7, BC = 2x + 1 in a right triangle configuration. Find x.

Solution

To solve this problem, we need to analyze the geometry of the triangle and the equations provided for each segment.

Given:

  • AB=x+2AB = x + 2
  • AD=x2AD = x - 2
  • DC=x+7DC = x + 7
  • BC=2x+1BC = 2x + 1

The diagram shows that ABD\triangle ABD and BDC\triangle BDC each contain right angles, with point BB as a shared vertex.

In a right triangle, the Pythagorean theorem applies: AB2=AD2+BD2andBC2=BD2+DC2AB^2 = AD^2 + BD^2 \quad \text{and} \quad BC^2 = BD^2 + DC^2 From these two equations, we can set up the following system:

  1. (x+2)2=(x2)2+BD2(x + 2)^2 = (x - 2)^2 + BD^2
  2. (2x+1)2=BD2+(x+7)2(2x + 1)^2 = BD^2 + (x + 7)^2

Step 1: Expand and Simplify Each Equation

  1. Expanding the terms in the first equation: (x+2)2=(x2)2+BD2(x + 2)^2 = (x - 2)^2 + BD^2 x2+4x+4=x24x+4+BD2x^2 + 4x + 4 = x^2 - 4x + 4 + BD^2 Simplifying, we get: 8x=BD28x = BD^2

  2. Expanding the terms in the second equation: (2x+1)2=BD2+(x+7)2(2x + 1)^2 = BD^2 + (x + 7)^2 4x2+4x+1=BD2+x2+14x+494x^2 + 4x + 1 = BD^2 + x^2 + 14x + 49 Substitute BD2=8xBD^2 = 8x from the first equation: 4x2+4x+1=8x+x2+14x+494x^2 + 4x + 1 = 8x + x^2 + 14x + 49 Simplifying further

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Right Triangles

Formulas

Pythagorean Theorem: a^2 + b^2 = c^2

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 8-10