Question

13: in ( \triangle wxy ), if ( angle x cong angle y ), ( wx = 9x - 11 ), ( xy = 4x + 1 ), and ( wy = 7x - 3 ), find ( x ) and the measure of each side.

Explanation:

Step1: Use the property of isosceles triangle

Since \(\angle X\cong\angle Y\), then \(WX = WY\) (sides opposite to congruent angles in a triangle are equal).
So, \(9x - 11=7x - 3\).

Step2: Solve the equation for \(x\)

Subtract \(7x\) from both sides: \(9x-7x - 11=7x - 7x- 3\), which gives \(2x-11=-3\).
Add \(11\) to both sides: \(2x-11 + 11=-3 + 11\), so \(2x = 8\).
Divide both sides by \(2\): \(x=\frac{8}{2}=4\).

Step3: Find the length of \(WX\)

Substitute \(x = 4\) into \(WX=9x - 11\). Then \(WX=9\times4-11=36 - 11 = 25\).

Step4: Find the length of \(WY\)

Substitute \(x = 4\) into \(WY = 7x-3\). Then \(WY=7\times4-3=28 - 3=25\).

Step5: Find the length of \(XY\)

Substitute \(x = 4\) into \(XY=4x + 1\). Then \(XY=4\times4+1=16 + 1=17\).

Answer:

\(x = 4\), \(WX = 25\), \(WY = 25\), \(XY=17\)