Question

if \\(\overline{ux} \cong \overline{wx}\\), \\(uv = p + 46\\), and \\(vw = 3p\\), what is \\(uv\\)?

Explanation:

Step1: Use the Hypotenuse - Leg (HL) Congruence Theorem

Since \(\overline{UX}\cong\overline{WX}\) and \(VX\perp UW\) (right angle), by the HL theorem, \(\triangle UVX\cong\triangle WVX\). Then \(UV = VW\).

Step2: Set up the equation

Given \(UV=p + 46\) and \(VW = 3p\), from \(UV=VW\), we have the equation \(p + 46=3p\).

Step3: Solve for \(p\)

Subtract \(p\) from both sides: \(46=3p - p\), so \(46 = 2p\). Then \(p=\frac{46}{2}=23\).

Step4: Find \(UV\)

Substitute \(p = 23\) into \(UV=p + 46\). So \(UV=23+46=70\).

Answer:

\(70\)