Question

1. given that \\( \overline{bd} \\) is the perpendicular bisector of \\( \overline{ca} \\) in the diagram below, find \\( ad \\).

\\( ad = 63 \\)

Explanation:

Step1: Recall Perpendicular Bisector Property

Since \( \overline{BD} \) is the perpendicular bisector of \( \overline{CA} \), by the Perpendicular Bisector Theorem, any point on the perpendicular bisector is equidistant from the endpoints of the segment. So, \( AD = CD \).

Step2: Set Up the Equation

We know \( CD = 8x + 7 \) and \( AD = 9x \). From the theorem, \( 9x = 8x + 7 \).

Step3: Solve for \( x \)

Subtract \( 8x \) from both sides: \( 9x - 8x = 8x + 7 - 8x \), which simplifies to \( x = 7 \).

Step4: Find \( AD \)

Substitute \( x = 7 \) into \( AD = 9x \): \( AD = 9 \times 7 = 63 \).

Answer:

\( AD = 63 \)