Question

given: right triangle abc with altitude \\(\overline{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} = \square\\). this can be rewritten as \\(\square = a^2\\). using ratios from the large and small triangles, \\(\square = \frac{b}{e}\\). this can be rewritten as \\(b^2 = ec\\). by substitution, \\(a^2 + b^2 = \square\\). you can then factor as \\(a^2 + b^2 = c(f + e)\\). from the large triangle, you know \\((f + e) = \square\\). so, \\(a^2 + b^2 = c^2\\) by using substitution.

Explanation:

Step1: Identify Similar Triangles

In right triangle \(ABC\) with altitude \(CD\), we have three similar triangles: \(\triangle ABC \sim \triangle CBD \sim \triangle ACD\). For the large triangle (\(\triangle ABC\)) and medium triangle (\(\triangle CBD\)), the ratio of corresponding sides gives \(\frac{c}{a}=\frac{a}{f}\) (since corresponding sides of similar triangles are proportional).

Step2: Rewrite the Ratio

Cross - multiplying \(\frac{c}{a}=\frac{a}{f}\) gives \(a^{2}=cf\).

Step3: Ratio for Large and Small Triangles

For the large triangle (\(\triangle ABC\)) and small triangle (\(\triangle ACD\)), the ratio of corresponding sides is \(\frac{c}{b}=\frac{b}{e}\) (by the property of similar triangles).

Step4: Rewrite the Second Ratio

Cross - multiplying \(\frac{c}{b}=\frac{b}{e}\) gives \(b^{2}=ec\).

Step5: Substitute into \(a^{2}+b^{2}\)

Substitute \(a^{2}=cf\) and \(b^{2}=ec\) into \(a^{2}+b^{2}\), we get \(a^{2}+b^{2}=cf + ec\).

Step6: Factor the Expression

Factor out \(c\) from \(cf+ec\), we have \(a^{2}+b^{2}=c(f + e)\).

Step7: Use the Length of the Hypotenuse

From the large triangle \(ABC\), the hypotenuse \(AB=c\) and \(AB = f + e\), so \(f + e=c\).

Step8: Final Substitution

Substitute \(f + e=c\) into \(a^{2}+b^{2}=c(f + e)\), we get \(a^{2}+b^{2}=c\times c=c^{2}\).

For the blanks:

• First blank (in \(\frac{c}{a}=\square\)): \(\frac{a}{f}\)
• Second blank (in \(\square=a^{2}\)): \(cf\)
• Third blank (in \(\square=\frac{b}{e}\)): \(\frac{c}{b}\)
• Fourth blank (in \(a^{2}+b^{2}=\square\)): \(cf + ec\)
• Fifth blank (in \((f + e)=\square\)): \(c\)

Answer:

First blank: \(\frac{a}{f}\); Second blank: \(cf\); Third blank: \(\frac{c}{b}\); Fourth blank: \(cf + ec\); Fifth blank: \(c\) (and the final proof shows \(a^{2}+b^{2}=c^{2}\))