Question

example 5 try it! identify and use the incenter of a triangle 5. use the figure shown. a. if m∠baf = 15 and m∠cbf = 52, what is m∠acf? b. if ef = 3y - 5 and df = 2y + 4, what is the distance from f to \\(\overline{ab}\\)?

Explanation:

Part (a)

Step1: Find \( m\angle BAC \)

Since \( F \) is the incenter, \( AF \) bisects \( \angle BAC \). Given \( m\angle BAF = 15^\circ \), so \( m\angle BAC = 2\times15^\circ = 30^\circ \).

Step2: Find \( m\angle ABC \)

\( BF \) bisects \( \angle ABC \), and \( m\angle CBF = 52^\circ \), so \( m\angle ABC = 2\times52^\circ = 104^\circ \).

Step3: Find \( m\angle ACB \)

In \( \triangle ABC \), the sum of angles is \( 180^\circ \). So \( m\angle ACB = 180^\circ - m\angle BAC - m\angle ABC = 180^\circ - 30^\circ - 104^\circ = 46^\circ \).

Step4: Find \( m\angle ACF \)

\( CF \) bisects \( \angle ACB \), so \( m\angle ACF = \frac{1}{2}m\angle ACB = \frac{1}{2}\times46^\circ = 23^\circ \).

Part (b)

Step1: Set \( EF = DF \) (Incenter property: distances to sides are equal)

Given \( EF = 3y - 5 \) and \( DF = 2y + 4 \), so \( 3y - 5 = 2y + 4 \).

Step2: Solve for \( y \)

Subtract \( 2y \) from both sides: \( 3y - 2y - 5 = 4 \), so \( y - 5 = 4 \). Add 5 to both sides: \( y = 9 \).

Step3: Find \( DF \) (distance from \( F \) to \( AB \))

Substitute \( y = 9 \) into \( DF = 2y + 4 \): \( DF = 2\times9 + 4 = 18 + 4 = 22 \). So the distance from \( F \) to \( \overline{AB} \) is \( 22 \).

Answer:

s:
a. \( \boldsymbol{23^\circ} \)
b. \( \boldsymbol{22} \)