Question

q is the midpoint of ru and t is the midpoint of su. if rs = b and qt = 4b - 91, what is qt? (there is a triangle - like figure with points s, r, u, t, q. t is on su, q is on ru, and qt is a segment.)

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Since \(Q\) is the midpoint of \(RU\) and \(P\) is the midpoint of \(SU\), then \(QP=\frac{1}{2}RS\).

Step2: Substitute the given values

We know that \(RS = \beta\) and \(QP = 4\beta-91\). Substituting into the equation \(QP=\frac{1}{2}RS\), we get \(4\beta - 91=\frac{1}{2}\beta\).

Step3: Solve the equation for \(\beta\)

First, multiply through by 2 to clear the fraction: \(2(4\beta - 91)=\beta\).
Expand: \(8\beta-182 = \beta\).
Subtract \(\beta\) from both sides: \(8\beta-\beta-182=\beta - \beta\), so \(7\beta-182 = 0\).
Add 182 to both sides: \(7\beta=182\).
Divide both sides by 7: \(\beta=\frac{182}{7}=26\).

Step4: Find the length of \(QP\)

Substitute \(\beta = 26\) into the expression for \(QP\): \(QP=4\beta-91\).
\(QP=4\times26 - 91\).
\(QP = 104-91\).
\(QP = 13\).

Answer:

\(13\)