Question

consider this right triangle with given measures. which of the following statements are true about the unknown leg length? select all that apply. the equation ((sqrt{34})^2 + 9^2 = c^2) will find the unknown leg length. the equation ((sqrt{34})^2 + b^2 = 9^2) will find the unknown leg length. the equation (a^2 + (sqrt{34})^2 = 9^2) will find the unknown leg length. the length of the unknown length is (sqrt{115}) m. the length of the unknown length is (sqrt{47}) m.

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is hypotenuse, \(a,b\) are legs. Here, hypotenuse is \(9\) m, one leg is \(\sqrt{34}\) m, let unknown leg be \(a\). So \(a^{2}+(\sqrt{34})^{2}=9^{2}\), or \((\sqrt{34})^{2}+b^{2}=9^{2}\) (letting unknown leg be \(b\)).

Step2: Analyze first statement

First statement: \((\sqrt{34})^{2}+9^{2}=c^{2}\). Here, \(9\) is hypotenuse, so this is wrong (hypotenuse should be \(c\), but \(9\) is hypotenuse, so sum of legs squared should equal hypotenuse squared, not leg plus hypotenuse squared).

Step3: Analyze second statement

Second statement: \((\sqrt{34})^{2}+b^{2}=9^{2}\). This is correct as per Pythagorean theorem (one leg \(\sqrt{34}\), unknown leg \(b\), hypotenuse \(9\)).

Step4: Analyze third statement

Third statement: \(a^{2}+(\sqrt{34})^{2}=9^{2}\). This is also correct (unknown leg \(a\), one leg \(\sqrt{34}\), hypotenuse \(9\)).

Step5: Calculate unknown leg length

From \(a^{2}+(\sqrt{34})^{2}=9^{2}\), \(a^{2}+ 34=81\), \(a^{2}=81 - 34=47\), so \(a = \sqrt{47}\) m. So fourth statement (\(\sqrt{115}\)) is wrong, fifth (\(\sqrt{47}\)) is correct.

Answer:

The true statements are:

• The equation \((\sqrt{34})^{2}+b^{2}=9^{2}\) will find the unknown leg length.
• The equation \(a^{2}+(\sqrt{34})^{2}=9^{2}\) will find the unknown leg length.
• The length of the unknown length is \(\sqrt{47}\) m.