Question

11 consider this right triangle

determine whether each expression can be used to find the length of side yz. mark yes or no for each expression.

Explanation:

Step1: Use Pythagorean theorem to find \(YZ\)

By Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 26\), \(a=10\), \(b = YZ\). So \(YZ=\sqrt{26^{2}-10^{2}}=\sqrt{(26 + 10)(26-10)}=\sqrt{36\times16}=24\)

Step2: Analyze \(26\sin(X)\)

In right - triangle \(\sin(X)=\frac{YZ}{XY}\), since \(XY = 26\), then \(26\sin(X)=YZ\)

Step3: Analyze \(\frac{10}{\tan(Y)}\)

In right - triangle \(\tan(Y)=\frac{XZ}{YZ}\), then \(YZ=\frac{XZ}{\tan(Y)}\), since \(XZ = 10\), so \(\frac{10}{\tan(Y)}=YZ\)

Step4: Analyze \(10\tan(X)\)

In right - triangle \(\tan(X)=\frac{YZ}{XZ}\), then \(YZ = 10\tan(X)\) is wrong. Because \(\tan(X)=\frac{YZ}{10}\), \(YZ = 10\tan(X)\)

Step5: Analyze \(\frac{10}{\tan(X)}\)

In right - triangle \(\tan(X)=\frac{YZ}{XZ}\), then \(\frac{10}{\tan(X)}=\frac{XZ}{\frac{YZ}{XZ}}=\frac{XZ^{2}}{YZ}
eq YZ\)

Step6: Analyze \(26\cos(Y)\)

In right - triangle \(\cos(Y)=\frac{YZ}{XY}\), since \(XY = 26\), then \(26\cos(Y)=YZ\)

Step7: Analyze \(\frac{26}{\cos(X)}\)

In right - triangle \(\cos(X)=\frac{XZ}{XY}\), then \(\frac{26}{\cos(X)}=\frac{XY}{\frac{XZ}{XY}}=\frac{XY^{2}}{XZ}
eq YZ\)

Answer:

Expression	Yes	No
\(\frac{10}{\tan(Y)}\)	Yes
\(10\tan(X)\)		Yes
\(\frac{10}{\tan(X)}\)		Yes
\(26\cos(Y)\)	Yes
\(\frac{26}{\cos(X)}\)		Yes