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Triangles

Long Answer Type

321. Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle.

Given: A ∆ABC in which medians drawn to the sides BC, CA and AB respectively.
To Prove : 3(AB² + BC² + CA²)
= 4(AD² + BE² + CF²)
Proof: We know that, the sum of the square of any two side is equal to twice the square of half of third side together with twice the square of median which bisects the third side.
Therefore,
Given: A ∆ABC in which medians drawn to the sides BC, CA and AB res

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322. Points P, Q, R and S in that order are dividing a line segment joining A(2, 6) and B(7, 4) in five equal parts. Find the co-ordinates of P and R.

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323. BL and CM are median of a triangle ABC right angled at A. Prove that 4(BL² + CM²) = 5BC².

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324.

In the given Fig, ABC is a right triangle, right angled at B. AD and CE are two medians drawn from A and C respectively. If AC = 5 cm and $AD equals fraction numerator 3 square root of 5 over denominator 2 end fraction cm comma$ find the length of CE.

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Short Answer Type

325. In an isosceles triangle ABC, with AB = AC, BD is perpendicular from B to the side AC. Prove that BD² - CD² = 2CD.AD.

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Long Answer Type

326. ABC is a triangle in which AB = AC and D is any point in BC. Prove that AB² - AD² = BD.CD.

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327. In a right triangle ABC, right angled at C.P. and Q are the points on the sides CA and CB respectively, which divdes these sides in the ratio 2:1. Prove that
(i) 9AQ² 9AC² + 4BC²(ii) 9BP² 9BC² + 4AC²(iii) 9(AQ² + BP²) = 13AB².

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