Question

follow these steps to derive the law of cosines. 1. the relationship between the side lengths in △abd is ( c^2 = x^2 + h^2 ) by the pythagorean theorem. 2. the relationship between the side lengths in △cbd is ( a^2 = (b - x)^2 + h^2 ) by the pythagorean theorem. 3. the equation ( a^2 = (b - x)^2 + h^2 ) is expanded to become ( a^2 = b^2 - 2bx + x^2 + h^2 ). 4. using the equation from step 1, the equation ( a^2 = b^2 - 2bx + x^2 + h^2 ) becomes ( a^2 = b^2 - 2bx + c^2 ) by dropdown with substitution, symmetry. (image: triangle abc with d, right angle at d, sides labeled x, b-x, b, c, a, h)

Explanation:

Step1: Recall Step 1's Equation

From Step 1, we have \( c^{2}=x^{2}+h^{2} \), which means \( x^{2}+h^{2}=c^{2} \).

Step2: Substitute into Step 3's Equation

In Step 3, the equation is \( a^{2}=b^{2}-2bx + x^{2}+h^{2} \). We substitute \( x^{2}+h^{2} \) with \( c^{2} \) (from Step 1's equation). So replacing \( x^{2}+h^{2} \) in \( a^{2}=b^{2}-2bx + x^{2}+h^{2} \) gives \( a^{2}=b^{2}-2bx + c^{2} \). This process is called substitution, as we substitute one expression (\( x^{2}+h^{2} \)) with its equivalent (\( c^{2} \)) from another equation.

Answer:

substitution