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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?

In ∆PQR and ∆PQR:
∠PST = ∠PST (Corresponding angles)
∠P = ∠P (Common)
Using AA similar condition.
$angle PST space tilde angle PQR$
∴ $fraction numerator ar open parentheses increment PQR close parentheses over denominator ar left parenthesis increment PST right parenthesis end fraction equals PQ squared over PS squared$
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$equals fraction numerator left parenthesis 2 plus 3 right parenthesis squared over denominator 3 squared end fraction equals 5 squared over 3 squared equals 25 over 9 space Ans. space$

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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.

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