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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.

Given: $increment ABC space$ in which $AD perpendicular BC.$
and $BD over DA equals DA over DC$
Given: in which and To Prove:∆ABC is a right triangleProof : I

To Prove:
∆ABC is a right triangle
Proof : In ∆ABD, we have
AB² = AD² + BD² ...(i)
[Using Pythagoras theorem]
In ∆ACD, we have
AC² = AD² + CD² ...(ii)
[Using Pythagoras theorem]
Adding (i) and (ii)
AB² + AC² = AD² + BD² + AD² + CD²⇒ AB² + AC² = 2AD² + BD² + CD²⇒ AB² + AC² = 2(BD.DC) + BD² + CD²⇒ AB² + AC² = (BD + DC)²⇒ AB² + AC² = BC²⇒ AB² + AC² = BC²Therefore, ∆ABC is a right triangle, right angled at A.

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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.

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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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