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In triangle ABC, AD is perpendicular to BC and BD is equal to one third of CD prove that 2 AC square is equal to 2AB square BC square?
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In triangle ABC, AD is perpendicular to BC and BD is equal to one thir...
In right ADB, AB²= AD² + BD² [By Pythagoras theorem] – (1)
In right ADC, AC²= AD² + CD² [By Pythagoras theorem] – (2)
Subtract (1) from (2),
we get AC² – AB2 = AD2 + CD2 – AD2 – BD2
AC2 – AB2 = CD2 – BD2 --- (3)
Given BD =1/3 CD
From the figure, BD + DC = BC
⇒ (1/3)CD + CD = BC
⇒ (4/3)CD = BC
∴ CD = 3BC/4
⇒ BD = BC/4
Xf Equation (3) becomes, AC2 – AB2 = (3BC/4)2 – (BC/4)2 = (8/16)BC2
⇒ AC2 – AB2 = (1/2) BC2
⇒ 2AC2 – 2AB2 = BC2
⇒ 2AC2 = 2AB2 + BC2
In right ADC, AC²= AD² + CD² [By Pythagoras theorem] – (2)
Subtract (1) from (2),
we get AC² – AB2 = AD2 + CD2 – AD2 – BD2
AC2 – AB2 = CD2 – BD2 --- (3)
Given BD =1/3 CD
From the figure, BD + DC = BC
⇒ (1/3)CD + CD = BC
⇒ (4/3)CD = BC
∴ CD = 3BC/4
⇒ BD = BC/4
Xf Equation (3) becomes, AC2 – AB2 = (3BC/4)2 – (BC/4)2 = (8/16)BC2
⇒ AC2 – AB2 = (1/2) BC2
⇒ 2AC2 – 2AB2 = BC2
⇒ 2AC2 = 2AB2 + BC2
Community Answer
In triangle ABC, AD is perpendicular to BC and BD is equal to one thir...
Proof:
1. Given:
- In triangle ABC, AD is perpendicular to BC.
- BD = 1/3 * CD
2. To prove:
- 2AC^2 = 2AB^2 + BC^2
3. Proof:
- Let's denote the lengths of the sides as follows:
- AC = x
- AB = y
- BC = z
4. By Pythagorean theorem:
- In triangle ABD, AD^2 + BD^2 = AB^2
- x^2 + (1/3 * z)^2 = y^2
- x^2 + (1/9)z^2 = y^2
5. By Pythagorean theorem:
- In triangle ACD, AD^2 + CD^2 = AC^2
- x^2 + z^2 = AC^2
6. Substitute the value of AC^2:
- x^2 + z^2 = 2AB^2 + BC^2
- 2AB^2 + BC^2 = 2AC^2
7. Hence, proved that:
- 2AC^2 = 2AB^2 + BC^2
8. Conclusion:
- The given relationship holds true in triangle ABC with the given conditions.
1. Given:
- In triangle ABC, AD is perpendicular to BC.
- BD = 1/3 * CD
2. To prove:
- 2AC^2 = 2AB^2 + BC^2
3. Proof:
- Let's denote the lengths of the sides as follows:
- AC = x
- AB = y
- BC = z
4. By Pythagorean theorem:
- In triangle ABD, AD^2 + BD^2 = AB^2
- x^2 + (1/3 * z)^2 = y^2
- x^2 + (1/9)z^2 = y^2
5. By Pythagorean theorem:
- In triangle ACD, AD^2 + CD^2 = AC^2
- x^2 + z^2 = AC^2
6. Substitute the value of AC^2:
- x^2 + z^2 = 2AB^2 + BC^2
- 2AB^2 + BC^2 = 2AC^2
7. Hence, proved that:
- 2AC^2 = 2AB^2 + BC^2
8. Conclusion:
- The given relationship holds true in triangle ABC with the given conditions.
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In triangle ABC, AD is perpendicular to BC and BD is equal to one third of CD prove that 2 AC square is equal to 2AB square BC square?
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In triangle ABC, AD is perpendicular to BC and BD is equal to one third of CD prove that 2 AC square is equal to 2AB square BC square? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In triangle ABC, AD is perpendicular to BC and BD is equal to one third of CD prove that 2 AC square is equal to 2AB square BC square? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In triangle ABC, AD is perpendicular to BC and BD is equal to one third of CD prove that 2 AC square is equal to 2AB square BC square?.
In triangle ABC, AD is perpendicular to BC and BD is equal to one third of CD prove that 2 AC square is equal to 2AB square BC square? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In triangle ABC, AD is perpendicular to BC and BD is equal to one third of CD prove that 2 AC square is equal to 2AB square BC square? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In triangle ABC, AD is perpendicular to BC and BD is equal to one third of CD prove that 2 AC square is equal to 2AB square BC square?.
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