Questions: In triangle ABC, the bisectors of angle A, angle B, and angle C cut the opposite sides onto lengths a1 and a2, b1 and b2, and c1 and c2, respectively, labeled in order counterclockwise around triangle ABC. Find the perimeter of triangle ABC for the set of values given below. a1 = 11/3, a2 = 22/3, b1 = 77/9 The perimeter is □

In triangle ABC, the bisectors of angle A, angle B, and angle C cut the opposite sides onto lengths a1 and a2, b1 and b2, and c1 and c2, respectively, labeled in order counterclockwise around triangle ABC. Find the perimeter of triangle ABC for the set of values given below.
a1 = 11/3, a2 = 22/3, b1 = 77/9

The perimeter is □

Transcript text: In $\triangle A B C$, the bisectors of $\angle \mathrm{A}, \angle \mathrm{B}$, and $\angle \mathrm{C}$ cut the opposite sides onto lengths $\mathrm{a}_{1}$ and $\mathrm{a}_{2}, \mathrm{~b}_{1}$ and $\mathrm{b}_{2}$, and $\mathrm{c}_{1}$ and $\mathrm{c}_{2}$, respectively, labeled in order counterclockwise around $\triangle A B C$. Find the perimeter of $\triangle A B C$ for the set of values given below. \[ a_{1}=\frac{11}{3}, a_{2}=\frac{22}{3}, b_{1}=\frac{77}{9} \] The perimeter is $\square$

Solution

Solution Steps

To find the perimeter of $\triangle ABC$, we need to determine the lengths of all sides $a$, $b$, and $c$. Given the lengths of segments created by the angle bisectors, we can use the Angle Bisector Theorem, which states that the ratio of the two segments created by the bisector is equal to the ratio of the other two sides of the triangle. Using this theorem, we can find the lengths of the sides and then sum them to get the perimeter.

Solution Approach

Use the Angle Bisector Theorem to find the lengths of sides $a$, $b$, and $c$.
Sum the lengths of the sides to find the perimeter.

Step 1: Given Values

We are given the lengths of the segments created by the angle bisectors in triangle $ABC$: \[ a_1 = \frac{11}{3}, \quad a_2 = \frac{22}{3}, \quad b_1 = \frac{77}{9} \]

Step 2: Calculate the Ratio $ \frac{b}{c} $

Using the Angle Bisector Theorem, we find the ratio of the segments: \[ \frac{a_1}{a_2} = \frac{b}{c} \] Calculating this gives: \[ \frac{b}{c} = \frac{11/3}{22/3} = \frac{1}{2} \]

Step 3: Calculate Length of Side $ b $

Let $ b_2 $ be the length of the other segment created by the bisector of angle $B$. From the ratio, we have: \[ b_1 + b_2 = b \] Using the ratio $ \frac{b}{c} = \frac{1}{2} $, we can express $ b_2 $ as: \[ b_2 = 2 \cdot b_1 = 2 \cdot \frac{77}{9} = \frac{154}{9} \] Thus, the length of side $ b $ is: \[ b = b_1 + b_2 = \frac{77}{9} + \frac{154}{9} = \frac{231}{9} \]

Step 4: Calculate Length of Side $ c $

Using the ratio $ \frac{b}{c} = \frac{1}{2} $, we can find $ c $: \[ c = 2b = 2 \cdot \frac{231}{9} = \frac{462}{9} \]

Step 5: Calculate Length of Side $ a $

The length of side $ a $ is simply the sum of the segments created by the bisector of angle $A$: \[ a = a_1 + a_2 = \frac{11}{3} + \frac{22}{3} = \frac{33}{3} = 11 \]

Step 6: Calculate the Perimeter

The perimeter $ P $ of triangle $ ABC $ is given by: \[ P = a + b + c = 11 + \frac{231}{9} + \frac{462}{9} \] Converting $ 11 $ to a fraction with a common denominator: \[ 11 = \frac{99}{9} \] Thus, the perimeter becomes: \[ P = \frac{99}{9} + \frac{231}{9} + \frac{462}{9} = \frac{99 + 231 + 462}{9} = \frac{792}{9} = 88 \]

Final Answer

The perimeter of triangle $ ABC $ is $\boxed{88}$.

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