Class 8 Exam > Class 8 Questions > Let ABC be a right angled triangle with angle...
Start Learning for Free
Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers?
Most Upvoted Answer
Let ABC be a right angled triangle with angle B 90°. Let the length of...
Understanding the Triangle Configuration
In triangle ABC, where angle B is 90°, BD is the altitude from B to AC. Given that BD = 12, we need to determine the minimum length of AC and ensure that the perimeter of triangle ABC consists of integers.
Area Calculation
- The area of triangle ABC can be expressed using the base AC and height BD:
- Area = 1/2 * AC * BD
- Substituting BD = 12:
- Area = 6 * AC
Relationship Between Sides
Using the properties of right triangles and the Pythagorean theorem:
- Let AB = c and BC = a.
- We have the following relationship:
- AC = sqrt(a^2 + c^2)
Using the area aspect:
- The area can also be represented as:
- Area = 1/2 * a * c
Equating the two area expressions gives us:
- 6 * AC = 1/2 * a * c
- This implies:
- AC = (a * c) / 12
Finding Integer Solutions
To minimize AC while ensuring the perimeter is an integer (P = a + c + AC):
1. Express AC in terms of integers:
- AC = (a * c) / 12
2. For AC to be minimized, a and c should be multiples of 12.
Assuming a = 12 and c = 12 (the smallest multiples):
- AC = (12 * 12) / 12 = 12
- Perimeter P = 12 + 12 + 12 = 36
Conclusion
Thus, the minimum possible length of AC is:
- AC = 12
And the perimeter of triangle ABC, ensuring all sides are integers, is:
- Perimeter = 36.
In triangle ABC, where angle B is 90°, BD is the altitude from B to AC. Given that BD = 12, we need to determine the minimum length of AC and ensure that the perimeter of triangle ABC consists of integers.
Area Calculation
- The area of triangle ABC can be expressed using the base AC and height BD:
- Area = 1/2 * AC * BD
- Substituting BD = 12:
- Area = 6 * AC
Relationship Between Sides
Using the properties of right triangles and the Pythagorean theorem:
- Let AB = c and BC = a.
- We have the following relationship:
- AC = sqrt(a^2 + c^2)
Using the area aspect:
- The area can also be represented as:
- Area = 1/2 * a * c
Equating the two area expressions gives us:
- 6 * AC = 1/2 * a * c
- This implies:
- AC = (a * c) / 12
Finding Integer Solutions
To minimize AC while ensuring the perimeter is an integer (P = a + c + AC):
1. Express AC in terms of integers:
- AC = (a * c) / 12
2. For AC to be minimized, a and c should be multiples of 12.
Assuming a = 12 and c = 12 (the smallest multiples):
- AC = (12 * 12) / 12 = 12
- Perimeter P = 12 + 12 + 12 = 36
Conclusion
Thus, the minimum possible length of AC is:
- AC = 12
And the perimeter of triangle ABC, ensuring all sides are integers, is:
- Perimeter = 36.
Attention Class 8 Students!
To make sure you are not studying endlessly, EduRev has designed Class 8 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 8.
|
Explore Courses for Class 8 exam
|
|
Similar Class 8 Doubts
Top Courses for Class 8View all
Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers?
Question Description
Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers?.
Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers?.
Solutions for Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers? in English & in Hindi are available as part of our courses for Class 8.
Download more important topics, notes, lectures and mock test series for Class 8 Exam by signing up for free.
Here you can find the meaning of Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers? defined & explained in the simplest way possible. Besides giving the explanation of
Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers?, a detailed solution for Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers? has been provided alongside types of Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers? theory, EduRev gives you an
ample number of questions to practice Let ABC be a right angled triangle with angle B 90°. Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC and the perimeter of triangle ABC are integers? tests, examples and also practice Class 8 tests.
|
|
Explore Courses for Class 8 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