Question

if ( st = p ) and ( ru = p - 33 ), what is the value of ( p )?

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment of a triangle (a segment connecting the mid - points of two sides of a triangle) is half the length of the third side. Here, \(RU\) is the mid - segment of \(\triangle STQ\) (since \(R\) is the mid - point of \(SQ\) and \(U\) is the mid - point of \(TQ\)), so \(RU=\frac{1}{2}ST\).

Step2: Substitute the given expressions

Given \(ST = p\) and \(RU=p - 33\), substituting into \(RU=\frac{1}{2}ST\) gives the equation \(p-33=\frac{1}{2}p\).

Step3: Solve the equation for \(p\)

Subtract \(\frac{1}{2}p\) from both sides: \(p-\frac{1}{2}p-33=\frac{1}{2}p-\frac{1}{2}p\), which simplifies to \(\frac{1}{2}p-33 = 0\). Then add 33 to both sides: \(\frac{1}{2}p-33 + 33=0 + 33\), so \(\frac{1}{2}p=33\). Multiply both sides by 2: \(p = 66\).

Answer:

\(66\)