Question

which proportional relationship is key to deriving the pythagorean theorem using triangle similarity?
a. ( ac = ab cdot ad ) and ( bc = ab cdot bd )
b. ( ac^2 = ab cdot ad ) and ( bc^2 = ab cdot bd )
c. ( ac^2 = ab cdot bc ) and ( bc^2 = ab cdot ad )
d. ( ac^2 = ab^2 ) and ( bc^2 = ad^2 )

Explanation:

To derive the Pythagorean Theorem using triangle similarity, we consider a right triangle \( ABC \) with a perpendicular \( CD \) (where \( D \) is on \( AB \)). Triangles \( ACD \sim ABC \) and \( BCD \sim ABC \) by AA similarity (right angle and common angle). For similar triangles, the ratio of corresponding sides is equal. For \( \triangle ACD \sim \triangle ABC \), \( \frac{AC}{AB}=\frac{AD}{AC} \), which gives \( AC^{2}=AB\cdot AD \). For \( \triangle BCD \sim \triangle ABC \), \( \frac{BC}{AB}=\frac{BD}{BC} \), which gives \( BC^{2}=AB\cdot BD \). Option a has linear terms, option c has incorrect products, and option d has incorrect equalities. So option b matches the proportional relationships from similar triangles.

Answer:

b. \( AC^{2} = AB \cdot AD \) and \( BC^{2} = AB \cdot BD \)