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 reason △acd ~ △cbd ~ △abc given $\frac{a}{c}=square,\frac{b}{c}=square$ corresponding sides of similar triangles are proportional. $a^{2}=cx,b^{2}=cy$ $a^{2}+b^{2}=cx + cy$ $a^{2}+b^{2}=c(x + y)$ distributive property $a^{2}+b^{2}=c^{2}$ $\frac{x}{a}$ $\frac{y}{b}$ $\frac{z}{a}$ $\frac{z}{b}$ addition property multiplication property substitution property pythagorean theorem

Explanation:

Step1: Identify proportional - sides

Since $\triangle ACD\sim\triangle ABC$, we have $\frac{a}{c}=\frac{x}{a}$; since $\triangle CBD\sim\triangle ABC$, we have $\frac{b}{c}=\frac{y}{b}$.

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 the two equations

Adding $a^{2}=cx$ and $b^{2}=cy$ gives $a^{2}+b^{2}=cx + cy$ (by addition property).

Step4: Factor out $c$

Using the distributive property, $cx+cy = c(x + y)$.

Step5: Substitute $x + y=c$

Since $x + y=c$ (from the figure, $AB=c=x + y$), we get $a^{2}+b^{2}=c^{2}$.

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, the reason for $a^{2}+b^{2}=c^{2}$ is substitution property.