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Prove that three times the sum of square of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle?
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Prove that three times the sum of square of the sides of a triangle is...
Apollonius theorem states that the sum of the squares of two sides of a triangle is equal to twice the square of the median on the third side plus half the square of the third side.
Hence AB2 + AC 2 = 2BD 2 + 2AD 2
= 2 � (�BC)2 + 2AD2
= � BC2 + 2AD2
∴ 2AB2 + 2AC 2 = BC2 + 4AD2 → (1)
Similarly, we get
2AB2 + 2BC2 = AC2 + 4BE2 → (2)
2BC2 + 2AC2 = AB2 + 4CF2 → (3)
Adding (1) (2) and (3), we get
4AB2 + 4BC2 + 4AC 2 = AB2 + BC2 + AC2 + 4AD2 + 4BE2 + 4CF2
3(AB2 + BC2 + AC2) = 4(AD2 + BE2 + CF2)
Hence, three times the sum of squares of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle.
Community Answer
Prove that three times the sum of square of the sides of a triangle is...
Proof:
Let ABC be a triangle with sides a, b, and c. Let G be the centroid of the triangle, and let ma, mb, and mc be the lengths of the medians from A, B, and C, respectively.
Step 1: Express the medians in terms of the sides.
We know that the median from A to BC has length ma = 1/2 * sqrt(2b^2 + 2c^2 - a^2). Similarly, we can find expressions for mb and mc in terms of a, b, and c.
Step 2: Square the expressions for the medians.
Squaring each of the expressions for the medians, we get:
ma2 = 1/4 * (2b^2 + 2c^2 - a^2)
mb2 = 1/4 * (2c^2 + 2a^2 - b^2)
mc2 = 1/4 * (2a^2 + 2b^2 - c^2)
Step 3: Add the squared medians.
Adding the squared medians, we get:
ma2 + mb2 + mc2 = 1/4 * (6a^2 + 6b^2 + 6c^2 - (a^2 + b^2 + c^2))
ma2 + mb2 + mc2 = 1/4 * (5a^2 + 5b^2 + 5c^2)
Step 4: Express the sum of the squares of the sides in terms of the medians.
We know that a^2 + b^2 + c^2 = 4/3 * (ma2 + mb2 + mc2 + 3/4 * (GA^2 + GB^2 + GC^2)), where GA, GB, and GC are the lengths of the segments from G to the vertices of the triangle.
Step 5: Substitute the expression for a^2 + b^2 + c^2 into the equation in Step 4.
Substituting the expression for a^2 + b^2 + c^2 into the equation in Step 4, we get:
3(a^2 + b^2 + c^2) = 4(ma2 + mb2
Let ABC be a triangle with sides a, b, and c. Let G be the centroid of the triangle, and let ma, mb, and mc be the lengths of the medians from A, B, and C, respectively.
Step 1: Express the medians in terms of the sides.
We know that the median from A to BC has length ma = 1/2 * sqrt(2b^2 + 2c^2 - a^2). Similarly, we can find expressions for mb and mc in terms of a, b, and c.
Step 2: Square the expressions for the medians.
Squaring each of the expressions for the medians, we get:
ma2 = 1/4 * (2b^2 + 2c^2 - a^2)
mb2 = 1/4 * (2c^2 + 2a^2 - b^2)
mc2 = 1/4 * (2a^2 + 2b^2 - c^2)
Step 3: Add the squared medians.
Adding the squared medians, we get:
ma2 + mb2 + mc2 = 1/4 * (6a^2 + 6b^2 + 6c^2 - (a^2 + b^2 + c^2))
ma2 + mb2 + mc2 = 1/4 * (5a^2 + 5b^2 + 5c^2)
Step 4: Express the sum of the squares of the sides in terms of the medians.
We know that a^2 + b^2 + c^2 = 4/3 * (ma2 + mb2 + mc2 + 3/4 * (GA^2 + GB^2 + GC^2)), where GA, GB, and GC are the lengths of the segments from G to the vertices of the triangle.
Step 5: Substitute the expression for a^2 + b^2 + c^2 into the equation in Step 4.
Substituting the expression for a^2 + b^2 + c^2 into the equation in Step 4, we get:
3(a^2 + b^2 + c^2) = 4(ma2 + mb2
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Prove that three times the sum of square of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle?
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Prove that three times the sum of square of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle? 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 Prove that three times the sum of square of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that three times the sum of square of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle?.
Prove that three times the sum of square of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle? 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 Prove that three times the sum of square of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that three times the sum of square of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle?.
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