Question

find each measure:

1. fg
2. tu

Explanation:

1. Solving for \(x\) and \(FG\) in the first triangle:

Step1: Set up the equation for \(x\)

Since \(EF = ED = 13\), by the Angle - Bisector Theorem (in a triangle, if a ray bisects an angle of the triangle and is perpendicular to the opposite side, then it bisects the opposite side). So \(FG=GD\), which gives \(5x - 17=3x + 1\).
Subtract \(3x\) from both sides: \(5x-3x - 17=3x - 3x+1\), so \(2x-17 = 1\).
Add \(17\) to both sides: \(2x-17 + 17=1 + 17\), then \(2x=18\).
Divide both sides by \(2\): \(x=\frac{18}{2}=9\).

Step2: Calculate \(FG\)

Substitute \(x = 9\) into the expression for \(FG\) (\(FG = 5x-17\)).
\(FG=5\times9-17=45 - 17=28\).

2. Solving for \(x\) and \(TU\) in the second triangle:

Step1: Set up the equation for \(x\)

Since \(ST = SR\) (marked as equal) and \(SU\perp TR\), by the property of isosceles triangles (the altitude from the vertex of an isosceles triangle to the base bisects the base). So \(TU = UR\), which gives \(2x + 24=5x-30\).
Subtract \(2x\) from both sides: \(2x-2x + 24=5x-2x-30\), so \(24 = 3x-30\).
Add \(30\) to both sides: \(24 + 30=3x-30 + 30\), then \(54=3x\).
Divide both sides by \(3\): \(x=\frac{54}{3}=18\).

Step2: Calculate \(TU\)

Substitute \(x = 18\) into the expression for \(TU\) (\(TU = 2x + 24\)).
\(TU=2\times18+24=36 + 24=60\).

Answer:

1. \(FG = 28\)
2. \(TU = 60\)