Question

find ac.

(image of a triangle with points b, d, e. segment bc = 8, cd = 8; segment ba = 10, ae = 10. segment ca is labeled 4w + 3, segment de is labeled 7w + 9. © algebra einstein, 2023)

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 midpoints of two sides of a triangle) is half the length of the third side. Here, \(CA\) is a mid - segment of \(\triangle BDE\) (since \(BC = CD = 8\) and \(BA=AE = 10\)). So, \(4w + 3=\frac{1}{2}(7w + 9)\).

Step2: Solve the equation for \(w\)

Multiply both sides of the equation \(4w + 3=\frac{1}{2}(7w + 9)\) by \(2\) to get \(2(4w + 3)=7w + 9\).
Expand the left - hand side: \(8w+6 = 7w + 9\).
Subtract \(7w\) from both sides: \(8w-7w+6=7w - 7w+9\), which gives \(w+6 = 9\).
Subtract \(6\) from both sides: \(w=9 - 6=3\).

Step3: Find the length of \(AC\)

Substitute \(w = 3\) into the expression for \(AC\) (\(AC=4w + 3\)).
\(AC=4\times3+3\).
\(AC = 12 + 3\).

Answer:

\(15\)