Question

given: right triangle abc with altitude cd
prove: ( a^{2}+b^{2}=c^{2} )
complete the paragraph proof.
you can use the similar triangles formed by the altitude to write ratios for corresponding sides. using ratios from the large and medium triangles, ( \frac{c}{a}=\frac{a}{f} ). this can be rewritten as ( f=\frac{a^{2}}{c} ). using ratios from the large and small triangles, ( \frac{c}{b}=\frac{b}{e} ). this can be rewritten as ( e=\frac{b^{2}}{c} ). by substitution, ( a^{2}+b^{2}=c f+c e ). you can then factor as ( a^{2}+b^{2}=c(f+e) ).
from the large triangle, you know ( (f+e)=c ). so, ( a^{2}+b^{2}=c^{2} ) by using substitution.

Explanation:

Step1: Use similarity of triangles

Since the triangles are similar, we can write ratios for corresponding sides. For the large and medium - sized triangles, \(\frac{a}{c}=\frac{f}{a}\). Cross - multiplying gives \(a^{2}=cf\).

Step2: Use similarity of another pair of triangles

For the large and small - sized triangles, \(\frac{b}{c}=\frac{e}{b}\). Cross - multiplying gives \(b^{2}=ce\).

Step3: Add the two equations

Adding the equations \(a^{2}=cf\) and \(b^{2}=ce\) gives \(a^{2}+b^{2}=cf + ce\). Factoring out \(c\) on the right - hand side, we get \(a^{2}+b^{2}=c(f + e)\).

Step4: Use the property of the hypotenuse

From the large triangle, we know that \(c=f + e\). Substituting \(c\) into the equation \(a^{2}+b^{2}=c(f + e)\), we have \(a^{2}+b^{2}=c\times c=c^{2}\).

Answer:

The blanks are filled as follows:

• First blank: \(\frac{a}{c}=\frac{f}{a}\), cross - multiply to get \(a^{2}=cf\)
• Second blank: \(\frac{b}{c}=\frac{e}{b}\), cross - multiply to get \(b^{2}=ce\)
• Third blank: \(a^{2}+b^{2}=cf+ce\)
• Fourth blank: \(c\) (because \(c = f + e\) from the large triangle)