+0

# help

0
69
2

Given that AB = 7.7, angle CAB = 36 degrees, BC = 9, and angle ACD = angle BCD.  Find AD.

Nov 18, 2019

#1
+2

Triangle ABC:
Using Law of Cosines:
Sides: AB = 7.7 BC = 9 AC = 14.009
Angle ∠ A = 30.191° = 30°11'27″ = 0.527 rad
Angle ∠ B =  36° = 0.628 rad
Angle ∠ C = 113.809° = 113°48'33″ = 1.986 rad

Using Law of Sines:
Sides: AD = 4.688 DC= 10.581 AC= 14.009
Angle ∠ C = 15.096° = 15°5'46″ = 0.263 rad
Angle ∠ A = 36° = 0.628 rad
Angle ∠ D = 128.904° = 128°54'14″ = 2.25 rad

Nov 18, 2019
#2
+106515
+1

Using the Law of Sines

sin 36 / 9  =  sin ACB / 7.7

sin ACB = 7.7 sin 36 / 9

arcsin (7.7 sin 36 / 9)  = ACB  ≈  30.19°

So angle  CBA = 180 - 36 - 30.19  ≈  113.81°

So....by the Law of sines

AC/ sin 113.81 = AB/ sin 30.19

AC / sin 113.81  =  7.7 / sin 30,19

AC = 7.7 sin13.81 / sin 30.19   ≈  14.0089

And because the apex angle is bisected

AC / AD  = BC / DB

AC / BC  =  AD/ DB

14.0089 / 9  = AD / DB

( 14.0089 / 9) DB = AD

(14.0089 / 9) DB + DB  = 7.7

(1 + 14.0089/9) DB = 7.7

DB =  7.7 / ( 1 + 14.0089/9) ≈   3.011

So

AD  = 7.7 - 3.011  ≈ 4.689

Nov 18, 2019
The text passage contains a geometry problem asking to find the length of AD given certain conditions about a triangle. Two answers are provided, one by a guest and the other by a user with a higher reputation, both using the Law of Sines and Law of Cosines to solve the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
+0

# help

0
69
2

Given that AB = 7.7, angle CAB = 36 degrees, BC = 9, and angle ACD = angle BCD.  Find AD.

Nov 18, 2019

#1
+2

Triangle ABC:
Using Law of Cosines:
Sides: AB = 7.7 BC = 9 AC = 14.009
Angle ∠ A = 30.191° = 30°11'27″ = 0.527 rad
Angle ∠ B =  36° = 0.628 rad
Angle ∠ C = 113.809° = 113°48'33″ = 1.986 rad

Using Law of Sines:
Sides: AD = 4.688 DC= 10.581 AC= 14.009
Angle ∠ C = 15.096° = 15°5'46″ = 0.263 rad
Angle ∠ A = 36° = 0.628 rad
Angle ∠ D = 128.904° = 128°54'14″ = 2.25 rad

Nov 18, 2019
#2
+106515
+1

Using the Law of Sines

sin 36 / 9  =  sin ACB / 7.7

sin ACB = 7.7 sin 36 / 9

arcsin (7.7 sin 36 / 9)  = ACB  ≈  30.19°

So angle  CBA = 180 - 36 - 30.19  ≈  113.81°

So....by the Law of sines

AC/ sin 113.81 = AB/ sin 30.19

AC / sin 113.81  =  7.7 / sin 30,19

AC = 7.7 sin13.81 / sin 30.19   ≈  14.0089

And because the apex angle is bisected

AC / AD  = BC / DB

AC / BC  =  AD/ DB

14.0089 / 9  = AD / DB

( 14.0089 / 9) DB = AD

(14.0089 / 9) DB + DB  = 7.7

(1 + 14.0089/9) DB = 7.7

DB =  7.7 / ( 1 + 14.0089/9) ≈   3.011

So

AD  = 7.7 - 3.011  ≈ 4.689

Nov 18, 2019
user
The text passage contains a geometry problem asking to find the length of AD given certain conditions about a triangle. Two answers are provided, one by a guest and the other by a user with a higher reputation, both using the Law of Sines and Law of Cosines to solve the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER