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Triangles

Long Answer Type

331.

In the given Fig. if AD ⊥ BC and $space BD over DA equals DA over DC.$ Prove that ∆ABC is a right triangle.

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Short Answer Type

332.

In the given Fig., AD ⊥ BC. Prove that AB² + CD² = BD² + AC².

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333. Triangle ABC is right-angled at B and D is the mid-point of BC. Prove that AC² = 4AD² - 3AB².

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334. In rhombus ABCD, prove that 4AB² = AC² + BD².

Given: A rhombus ABCD
To Prove: 4AB² = AC² + BD²Proof: The diagonals of a rhombus bisect each other at right angles.
Given: A rhombus ABCDTo Prove: 4AB2 = AC2 + BD2Proof: The diagonals
∴ $space space OA equals 1 half AC comma space OB equals 1 half BD$
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Now, in right $increment AOB comma space$ using the above theorem,
$space AB squared space equals space OA squared plus OB squared space space space space space space space space equals space open parentheses 1 half AC close parentheses squared plus open parentheses 1 half BD close parentheses squared space space space space space space space space space equals space AC squared plus BD squared Hence comma space space 4 AB squared space equals space AC squared plus BD squared space space space space space Proved.$

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335.

In the given Fig., DE || BC, if $AD over DB equals 3 over 2$ and AE = 4.8 cm, find EC.

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Long Answer Type

336. In the given figure, PQ || BC. If
(i)

AP over PB equals 2 over 3 space and space AC space equals space 16 space cm comma space space space find space QC.

(ii)

AQ over CQ equals 3 over 5 space and space PB space equals space 7.5 space cm comma space find space AB.

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337. If three or more parallel lines are intersected by two transversals. Prove that the intercepts made by them on the transversals are proportional.

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Short Answer Type

338. In quadrilateral ABCD, P is any point in side BC. PQ || AB and PR || BD. Show that QR is parallel to AD.

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339. If D and E are respectively, the points on the sides AB and AC of a triangle ABC, such that AD = 6 cm, BD = 9 cm, AE = 8 cm and EC = 12 cm then show that DE || BC.

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