Question

the figure shows right triangle abc with altitude cd, where △acd ~ △cbd ~ △abc. move the options to the spaces to prove the pythagorean theorem.
statement\treason
△acd ~ △cbd ~ △abc\tgiven
\frac{a}{c}=square,\frac{b}{c}=square\tcorresponding sides of similar triangles are proportional.
a² = cx, b² = cy

a² + b² = cx + cy

a² + b² = c(x + y)\tdistributive property
a² + b² = c²

\frac{x}{a}\frac{y}{b}\frac{z}{a}\frac{z}{b} addition property multiplication property substitution property pythagorean theorem

Explanation:

Step1: Identify similar - triangle ratios

Since $\triangle ACD\sim\triangle ABC$ and $\triangle CBD\sim\triangle ABC$, for $\frac{a}{c}$, the corresponding side ratio gives $\frac{a}{c}=\frac{x}{a}$; for $\frac{b}{c}$, the corresponding side ratio gives $\frac{b}{c}=\frac{y}{b}$.

Step2: Cross - multiply

Cross - multiplying $\frac{a}{c}=\frac{x}{a}$ gives $a^{2}=cx$, and cross - multiplying $\frac{b}{c}=\frac{y}{b}$ gives $b^{2}=cy$.

Step3: Add equations

Adding $a^{2}=cx$ and $b^{2}=cy$ gives $a^{2}+b^{2}=cx + cy$ by the addition property of equality.

Step4: Apply distributive property

Factor out $c$ from the right - hand side: $a^{2}+b^{2}=c(x + y)$.

Step5: Use the fact that $x + y=c$

Since $x + y$ is the length of the hypotenuse $c$, we get $a^{2}+b^{2}=c^{2}$ by substitution property.

Answer:

The first blank is $\frac{x}{a}$, the second blank is $\frac{y}{b}$, the reason for $a^{2}=cx,b^{2}=cy$ is multiplication property, the reason for $a^{2}+b^{2}=cx + cy$ is addition property, and the reason for $a^{2}+b^{2}=c^{2}$ is substitution property.