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Triangles

Long Answer Type

251.

In the given Fig, two triangles ABC and DBC lie on the same side of base BC. P is a point on BC such that PQ || BA and PR || BD. Prove that: QR || AD.

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Short Answer Type

252.

In the given fig, if ∠P = ∠RTS. Prove that ∆RPQ ~ ∆RTS.

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253. In a ∆ABC, DE || BC, if

DE equals 2 over 3 BC

and area of ∆ABC = 81 cm². Find the area of ∆ADE.

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254.

In the given fig, ST || QR, PS = 2 cm and SQ = 3 cm. What is the ratio of the area of ∆PQR to the area of ∆PST?

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255. Prove that the diagonals of a rectangle bisect each other and are equal.

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256.

In the given Fig. PB and QA are perpendiculars to segment AB. If PO = 5 cm, QO = 7 cm and area ∆POB = 150 cm² find the area of ∆QOA.

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257. In two similar triangles ABC and PQR, if their corresponding altitudes AD and PS are in the ratio 4 : 9, find the ratio of the areas of ∆ABC and ∆PQR.

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258. In ∆ABC shown in Fig, DE || BC. If BC = 8 cm, DE = 6 cm and area of ∆ADE = 45 cm², what is the area of ∆ABC.

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Long Answer Type

259. If D is a point on the side AB of ∆ABC such that AD : DB = 3:2 and E is a point on BC such that DE || AC ; find the ratio of the areas of ∆ABC and ∆BDE.

In ∆BDE and ∆ABC
∠B = ∠B    [Common]
∴    ∠1 = ∠2 [Correspondeing ∠S]
Now, ∆BDE ~ ∆ABC
∴    $fraction numerator ar left parenthesis increment BDE right parenthesis over denominator ar left parenthesis increment ABC right parenthesis end fraction equals BD squared over AB squared$

We have, $AD over DB equals 3 over 2$

$rightwards double arrow$ $AD over DB plus 1 equals 3 over 2 plus 1$

$rightwards double arrow$    $fraction numerator AD plus DB over denominator DB end fraction equals 5 over 2$

$rightwards double arrow$    $AB over DB equals 5 over 2 space or comma space DB over AB equals 2 over 5 space space space... left parenthesis ii right parenthesis$

In ∆BDE and ∆ABC∠B = ∠B [Common]∴ ∠1 = ∠2 [

Putting (ii) in (i), we get
$fraction numerator ar left parenthesis increment BDE right parenthesis over denominator ar left parenthesis increment ABC right parenthesis end fraction equals 4 over 25$

$rightwards double arrow$ $fraction numerator ar left parenthesis increment ABC right parenthesis over denominator ar left parenthesis increment BDE right parenthesis end fraction equals 25 over 4$
(Taking reciprocals of both side)

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Short Answer Type

260.

In the given fig, DE is parallel to BC and AD = 1 cm, BD = 2 cm. What is the ratio of the area of ∆ABC to the area of ∆ADE?

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