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Mathematics : Triangles

Long Answer Type

401. D,E and F are points on BC, CA and AB respectively of ∆ABC, such that AD bisects ∠A, BE bisects ∠B and CF bisects ∠C. If AB = 5 cm, BC = 8 cm and CA = 4 cm, determine AF, CE and BD.

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402. In ∆ABC, ∠A = 90° and AD ⊥ BC. If AB = 12 cm and the area of ∆ABC = 30 cm², find AD.

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403.

In the Fig., ABC is a triangle in which AB = AC points D and E are points on the sides AB and AC respectively such that AD = AE. Show that the points B, C, E and D are concyclic.

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404. ABC is an isoscles triangle right angled at B. Two equilateral triangles are constructed with sides BC and AC as shown in Fig. Prove the area of ∆BCD =

1 half area space of space DACF.

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405. L and M are the mid-points of AB and respectively of ∆ABC, right-angled at B. Prove that 4LC² = AB² + 4BC².

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406.

In an equilateral triangle ABC, AD is the altitude drawn from A on side BC. Prove that:
3AB² = 4AD².

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407. Two opposite sides AB and DC of a quadrilateral ABCD intersect at E, when produced. Prove that triangles EAD and ECB are similar.

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Short Answer Type

408. ABC is a triangle PQ is the line segment intersecting AB in P and AC in Q such that PQ || BC and divides ∆ABC into two parts equal in area. Find BP:AB.

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409. In an equilateral triangle with side a, prove that the area of the triangle is

fraction numerator straight a squared square root of 3 over denominator 4 end fraction.

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Long Answer Type

410. Prove that the equilateral triangles described on the two sides of a right-angled triangle are together equal to equilateral triangle on the hypotenuse in terms of their areas.

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