Question

1. use the figure shown.

a. if ( mangle baf = 15 ) and ( mangle cbf = 52 ), what is ( mangle acf )?
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b. if ( ef = 3y - 5 ) and ( df = 2y + 4 ), what is the distance from ( f ) to ( overline{ab} )?
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Explanation:

a.

Step1: Use the Angle - Bisector Theorem (Exterior Angle Property)

By the exterior angle property of a triangle, \(m\angle ACF=m\angle BAF + m\angle CBF\).

Step2: Substitute the given values

Given \(m\angle BAF = 15^{\circ}\) and \(m\angle CBF=52^{\circ}\), then \(m\angle ACF=15 + 52\).

b.

Step1: Use the property of the angle - bisector (distance from a point to the sides of an angle)

Since \(F\) is on the angle - bisector, \(EF = DF\). So, \(3y-5=2y + 4\).

Step2: Solve for \(y\)

Subtract \(2y\) from both sides: \(3y-2y-5=2y-2y + 4\), which gives \(y-5 = 4\). Then add \(5\) to both sides: \(y=4 + 5=9\).

Step3: Find the length of \(EF\) (or \(DF\))

Substitute \(y = 9\) into \(EF=3y-5\). So, \(EF=3\times9-5=27-5 = 22\). The distance from \(F\) to \(\overline{AB}\) is equal to \(EF\) (or \(DF\)) because of the angle - bisector property.

Answer:

a. \(67\)
b. \(22\)