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If in triangle ABC, AD is a median and AM is perpendicular to BC, then prove that AB2 = AD2-BC*DM + 1/4 BC2?
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If in triangle ABC, AD is a median and AM is perpendicular to BC, then...
Solution:-
D is the midpoint of BC and AM ⊥ BC.
In right angled triangle ABM,
AB² = AM² + BM² ....(1) - Pythagoras Theorem
In right angled triangle ADM,
AD² = AM² + MD² ....(2) - Pythagoras Theorem
From (1) and (2), we get
AB² = AD² - MD² + BM²
⇒ AB² = AD² - DM² + (BD - DM)²
⇒ AB² = AD² - DM² + BD² + DM² - 2BD × DM
⇒ AB² = AD² - 2BD × DM + BD²
⇒ AB² = AD² - 2(BC/2) × DM + (BC/2)² {∵ BD = DC = BC/2}
⇒ AB² = AD² - BC × DM + BC²/4
Hence proved.
D is the midpoint of BC and AM ⊥ BC.
In right angled triangle ABM,
AB² = AM² + BM² ....(1) - Pythagoras Theorem
In right angled triangle ADM,
AD² = AM² + MD² ....(2) - Pythagoras Theorem
From (1) and (2), we get
AB² = AD² - MD² + BM²
⇒ AB² = AD² - DM² + (BD - DM)²
⇒ AB² = AD² - DM² + BD² + DM² - 2BD × DM
⇒ AB² = AD² - 2BD × DM + BD²
⇒ AB² = AD² - 2(BC/2) × DM + (BC/2)² {∵ BD = DC = BC/2}
⇒ AB² = AD² - BC × DM + BC²/4
Hence proved.
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If in triangle ABC, AD is a median and AM is perpendicular to BC, then prove that AB2 = AD2-BC*DM + 1/4 BC2?
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If in triangle ABC, AD is a median and AM is perpendicular to BC, then prove that AB2 = AD2-BC*DM + 1/4 BC2? 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 If in triangle ABC, AD is a median and AM is perpendicular to BC, then prove that AB2 = AD2-BC*DM + 1/4 BC2? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If in triangle ABC, AD is a median and AM is perpendicular to BC, then prove that AB2 = AD2-BC*DM + 1/4 BC2?.
If in triangle ABC, AD is a median and AM is perpendicular to BC, then prove that AB2 = AD2-BC*DM + 1/4 BC2? 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 If in triangle ABC, AD is a median and AM is perpendicular to BC, then prove that AB2 = AD2-BC*DM + 1/4 BC2? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If in triangle ABC, AD is a median and AM is perpendicular to BC, then prove that AB2 = AD2-BC*DM + 1/4 BC2?.
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