the figure shows right triangle abc with altitude $\overline{cd}$, where $\triangle acd \sim \triangle cbd \sim \triangle abc$.
move the options to the spaces to prove the pythagorean theorem.
| statement | reason |
| --- | --- |
| $\triangle acd \sim \triangle cbd \sim \triangle abc$ | 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 |
| $a^2 + b^2 = c^2$ | |
$\frac{x}{a}$ $\frac{y}{b}$ $\frac{c}{a}$ $\frac{c}{b}$ addition property multiplication property substitution property pythagorean theorem

Question

the figure shows right triangle abc with altitude $\overline{cd}$, where $\triangle acd \sim \triangle cbd \sim \triangle abc$.
move the options to the spaces to prove the pythagorean theorem.
| statement | reason |
| --- | --- |
| $\triangle acd \sim \triangle cbd \sim \triangle abc$ | 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 |
| $a^2 + b^2 = c^2$ |  |
$\frac{x}{a}$ $\frac{y}{b}$ $\frac{c}{a}$ $\frac{c}{b}$ addition property multiplication property substitution property pythagorean theorem

Accepted Answer

# Explanation:
## Step1: Fill proportionality blanks
From similar triangles $	riangle CBD \sim 	riangle ABC$, $\frac{a}{c}=\frac{x}{a}$. From $	riangle ACD \sim 	riangle ABC$, $\frac{b}{c}=\frac{y}{b}$.
## Step2: Justify cross-multiplication
Cross-multiply the proportions: $\frac{a}{c}=\frac{x}{a} \implies a^2 = cx$; $\frac{b}{c}=\frac{y}{b} \implies b^2 = cy$. Reason: Multiplication property.
## Step3: Justify adding equations
Add $a^2 = cx$ and $b^2 = cy$. Reason: Addition property.
## Step4: Justify final substitution
Since $x + y = c$ (segment addition on hypotenuse $AB$), substitute into $a^2 + b^2 = c(x + y)$. Reason: Substitution property.

# Answer:
| Statement | Reason |
|-----------|--------|
| $	riangle ACD \sim 	riangle CBD \sim 	riangle ABC$ | Given |
| $\frac{a}{c}=\boldsymbol{\frac{x}{a}}$, $\frac{b}{c}=\boldsymbol{\frac{y}{b}}$ | Corresponding sides of similar triangles are proportional. |
| $a^2 = cx, b^2 = cy$ | **Multiplication property** |
| $a^2 + b^2 = cx + cy$ | **Addition property** |
| $a^2 + b^2 = c(x + y)$ | Distributive property |
| $a^2 + b^2 = c^2$ | **Substitution property** |

statement	reason
\\(\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\\)

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QUESTION IMAGE

Question

Explanation:

Step1: Fill proportionality blanks

Step2: Justify cross-multiplication

Step3: Justify adding equations

Step4: Justify final substitution

Answer:

Statement	Reason
$\frac{a}{c}=\boldsymbol{\frac{x}{a}}$, $\frac{b}{c}=\boldsymbol{\frac{y}{b}}$	Corresponding sides of similar triangles are proportional.
$a^2 = cx, b^2 = cy$	Multiplication property
$a^2 + b^2 = cx + cy$	Addition property
$a^2 + b^2 = c(x + y)$	Distributive property
$a^2 + b^2 = c^2$	Substitution property