Question

1. given (overline{mh}) as shown with (overline{mt} cong overline{ah}), (mt = 5x - 8) and (ah = 3x + 12). if (at = 6), find (th).

Explanation:

Step1: Set MT equal to AH (since MT ≅ AH)

Since \( \overline{MT} \cong \overline{AH} \), their lengths are equal. So we set up the equation:
\( 5x - 8 = 3x + 12 \)

Step2: Solve for x

Subtract \( 3x \) from both sides:
\( 5x - 3x - 8 = 12 \)
\( 2x - 8 = 12 \)

Add 8 to both sides:
\( 2x = 12 + 8 \)
\( 2x = 20 \)

Divide both sides by 2:
\( x = \frac{20}{2} = 10 \)

Step3: Find the length of AH (or MT, since they are congruent)

Substitute \( x = 10 \) into the expression for \( AH \) (or \( MT \)):
\( AH = 3(10) + 12 = 30 + 12 = 42 \)

Step4: Find TH

We know \( AT = 6 \) and \( AH = AT + TH \) (from the segment addition postulate, since A, T, H are colinear with T between A and H). So:
\( TH = AH - AT \)
Substitute \( AH = 42 \) and \( AT = 6 \):
\( TH = 42 - 6 = 36 \)

Answer:

\( 36 \)