JEE Exam > JEE Questions > In a triangle ABC angles are in arithmetic pr...
Start Learning for Free
In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16?
Most Upvoted Answer
In a triangle ABC angles are in arithmetic progression with common dif...
Given:
- Triangle ABC with angles in arithmetic progression with common difference α.
- cos(α) = 21/22.
- The triangle is inscribed in a circle with radius 11 units.
- H is the orthocenter of the triangle.
To find:
The value of HA * HB * HC / 16.
Solution:
Step 1: Determine the measure of each angle in the triangle.
Let the angles of the triangle be A, A + α, and A + 2α.
Using the property that the sum of the angles in a triangle is 180 degrees, we have:
A + A + α + A + 2α = 180
3A + 3α = 180
A + α = 60
Since the angles are in an arithmetic progression, we have:
A + 2α - (A + α) = α = 60 - A
Step 2: Determine the values of A and α.
Using the given value of cos(α) = 21/22, we can find the value of sin(α) using the Pythagorean identity:
sin^2(α) + cos^2(α) = 1
sin^2(α) = 1 - cos^2(α)
sin^2(α) = 1 - (21/22)^2
sin^2(α) = 484/484 - 441/484
sin^2(α) = 43/484
sin(α) = √(43/484)
sin(α) = √43 / 22
We know that sin(α) = sin(π/2 - α). Therefore, sin(π/2 - α) = √43 / 22.
Using the identity sin(π/2 - α) = √(1 - cos^2(α)), we have:
√(1 - cos^2(α)) = √43 / 22
1 - cos^2(α) = 43 / 484
cos^2(α) = 441 / 484
cos(α) = ± 21 / 22
Since α is acute, we take the positive value: cos(α) = 21 / 22.
Now, we have cos(α) = 21 / 22, and α = 60 - A.
Substituting these values into the equation cos(α) = 21 / 22, we get:
cos(60 - A) = 21 / 22
Using the property cos(60 - A) = cos(A - 60), we can rewrite the equation as:
cos(A - 60) = 21 / 22
Step 3: Solve for A.
To solve this equation, we can use the inverse cosine function:
A - 60 = arccos(21 / 22)
A = arccos(21 / 22) + 60
Using a calculator, we find A ≈ 18.02 degrees.
Step 4: Calculate the measures of each angle.
Since A + α = 60, we have:
18.02 + α = 60
α = 60 - 18.02
- Triangle ABC with angles in arithmetic progression with common difference α.
- cos(α) = 21/22.
- The triangle is inscribed in a circle with radius 11 units.
- H is the orthocenter of the triangle.
To find:
The value of HA * HB * HC / 16.
Solution:
Step 1: Determine the measure of each angle in the triangle.
Let the angles of the triangle be A, A + α, and A + 2α.
Using the property that the sum of the angles in a triangle is 180 degrees, we have:
A + A + α + A + 2α = 180
3A + 3α = 180
A + α = 60
Since the angles are in an arithmetic progression, we have:
A + 2α - (A + α) = α = 60 - A
Step 2: Determine the values of A and α.
Using the given value of cos(α) = 21/22, we can find the value of sin(α) using the Pythagorean identity:
sin^2(α) + cos^2(α) = 1
sin^2(α) = 1 - cos^2(α)
sin^2(α) = 1 - (21/22)^2
sin^2(α) = 484/484 - 441/484
sin^2(α) = 43/484
sin(α) = √(43/484)
sin(α) = √43 / 22
We know that sin(α) = sin(π/2 - α). Therefore, sin(π/2 - α) = √43 / 22.
Using the identity sin(π/2 - α) = √(1 - cos^2(α)), we have:
√(1 - cos^2(α)) = √43 / 22
1 - cos^2(α) = 43 / 484
cos^2(α) = 441 / 484
cos(α) = ± 21 / 22
Since α is acute, we take the positive value: cos(α) = 21 / 22.
Now, we have cos(α) = 21 / 22, and α = 60 - A.
Substituting these values into the equation cos(α) = 21 / 22, we get:
cos(60 - A) = 21 / 22
Using the property cos(60 - A) = cos(A - 60), we can rewrite the equation as:
cos(A - 60) = 21 / 22
Step 3: Solve for A.
To solve this equation, we can use the inverse cosine function:
A - 60 = arccos(21 / 22)
A = arccos(21 / 22) + 60
Using a calculator, we find A ≈ 18.02 degrees.
Step 4: Calculate the measures of each angle.
Since A + α = 60, we have:
18.02 + α = 60
α = 60 - 18.02
Community Answer
In a triangle ABC angles are in arithmetic progression with common dif...
2
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16?
Question Description
In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16?.
In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16?.
Solutions for In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16? defined & explained in the simplest way possible. Besides giving the explanation of
In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16?, a detailed solution for In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16? has been provided alongside types of In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16? theory, EduRev gives you an
ample number of questions to practice In a triangle ABC angles are in arithmetic progression with common difference alpha such that cos( alpha) is equal to 21/22 and the triangle is inscribed in a circle with radius 11 units and H is orthocentre of the triangle then find HA HB HC / 16? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup