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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.

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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?

It is given that AD = 1 cm, BD = 2 cm and DE || BC
In ∆ADE and ∆ABC,
∠ADE = ∠ABC (Corresponding angles)
∠A = ∠A [Common]
Therefore, by A.A. similar condition
∆ADE ~ ∆ABC
Ratio of areas of similar triangles is equal to the square of the ratio of the corresponding sides.
∴ $It is given that AD = 1 cm, BD = 2 cm and DE || BCIn ∆ADE and ∆AB$ $fraction numerator ar left parenthesis increment ABC right parenthesis over denominator ar left parenthesis increment ADE right parenthesis end fraction equals AB squared over AD squared$
$rightwards double arrow space space space space space space space fraction numerator ar left parenthesis increment ABC right parenthesis over denominator ar left parenthesis increment ADE right parenthesis end fraction equals open parentheses 3 over 1 close parentheses squared rightwards double arrow space space space space space space space space space fraction numerator ar left parenthesis increment ABC right parenthesis over denominator ar left parenthesis increment ADE right parenthesis end fraction space equals space 9 over 1.$