Class 10 Exam > Class 10 Questions > In an equilateral triangle ABC, ifAD⊥BC,...
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In an equilateral triangle ABC, if AD⊥BC,then
- a)5AB2 = 4AD2.
- b)4AB2 = 3AD2.
- c)3AB2 = 4AD2.
- d)2AB2 = 3AD2.
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In an equilateral triangle ABC, ifAD⊥BC,thena)5AB2 = 4AD2.b)4AB2 ...
Since the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.
Most Upvoted Answer
In an equilateral triangle ABC, ifAD⊥BC,thena)5AB2 = 4AD2.b)4AB2 ...
In an equilateral triangle ABC, if AD is the perpendicular bisector of BC, then D is the midpoint of BC.
Since ABC is an equilateral triangle, all sides are equal in length. Let's assume the length of side AB (or BC) is "x".
Since AD is the perpendicular bisector of BC, it divides BC into two equal segments. Let's call the length of BD and CD as "y".
Since D is the midpoint of BC, BD = CD = y.
Now, let's consider triangle ADB. It is a right triangle with AD as the hypotenuse and BD as one of the legs. By the Pythagorean theorem, we have:
AD^2 = AB^2 - BD^2
Substituting the values, we get:
AD^2 = x^2 - y^2
Since AB = BC = x, we can rewrite it as:
AD^2 = x^2 - y^2
Since AD = DC = x/2 (as AD is the perpendicular bisector), we can rewrite it as:
(x/2)^2 = x^2 - y^2
Simplifying, we get:
x^2/4 = x^2 - y^2
Multiplying both sides by 4, we get:
x^2 = 4x^2 - 4y^2
Simplifying further, we get:
3x^2 = 4y^2
Dividing both sides by 4, we get:
3/4 * x^2 = y^2
Taking the square root of both sides, we get:
sqrt(3/4 * x^2) = y
Simplifying, we get:
sqrt(3)/2 * x = y
Since BD = CD = y, we can conclude that D is the midpoint of BC.
Since ABC is an equilateral triangle, all sides are equal in length. Let's assume the length of side AB (or BC) is "x".
Since AD is the perpendicular bisector of BC, it divides BC into two equal segments. Let's call the length of BD and CD as "y".
Since D is the midpoint of BC, BD = CD = y.
Now, let's consider triangle ADB. It is a right triangle with AD as the hypotenuse and BD as one of the legs. By the Pythagorean theorem, we have:
AD^2 = AB^2 - BD^2
Substituting the values, we get:
AD^2 = x^2 - y^2
Since AB = BC = x, we can rewrite it as:
AD^2 = x^2 - y^2
Since AD = DC = x/2 (as AD is the perpendicular bisector), we can rewrite it as:
(x/2)^2 = x^2 - y^2
Simplifying, we get:
x^2/4 = x^2 - y^2
Multiplying both sides by 4, we get:
x^2 = 4x^2 - 4y^2
Simplifying further, we get:
3x^2 = 4y^2
Dividing both sides by 4, we get:
3/4 * x^2 = y^2
Taking the square root of both sides, we get:
sqrt(3/4 * x^2) = y
Simplifying, we get:
sqrt(3)/2 * x = y
Since BD = CD = y, we can conclude that D is the midpoint of BC.
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In an equilateral triangle ABC, ifAD⊥BC,thena)5AB2 = 4AD2.b)4AB2 = 3AD2.c)3AB2 = 4AD2.d)2AB2 = 3AD2.Correct answer is option 'C'. Can you explain this answer?
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