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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.

In ∆ADE,
DE || BC (Given)
∴ ∠D = ∠B, ∠E = ∠C [corres. ∠s]
and ∠A = ∠A [Common]
∴ ∆ADE ~ ∆ABC
[A.A.A. Similarity]
$rightwards double arrow$ $fraction numerator ar left parenthesis increment ADE right parenthesis over denominator ar left parenthesis increment ABC right parenthesis end fraction equals DE squared over BC squared equals fraction numerator left parenthesis 6 right parenthesis squared over denominator left parenthesis 8 right parenthesis squared end fraction$
$rightwards double arrow$ $space space fraction numerator 45 space cm squared over denominator ar left parenthesis increment ABC right parenthesis end fraction equals 36 over 64$

$rightwards double arrow$ $space space space space space space ar left parenthesis increment ABC right parenthesis space equals space fraction numerator 45 cross times 64 over denominator 36 end fraction equals 80 space cm squared$

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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.

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