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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.

DE over BC equals 2 over 3 space space space space space space space space space space space space left parenthesis Given right parenthesis

DE space parallel to space BC

(Given)

rightwards double arrow

by AA similarity

increment ADE tilde increment ABC

By area ratio theorem

fraction numerator ar left parenthesis increment ADE right parenthesis over denominator ar left parenthesis increment ABC right parenthesis end fraction equals open parentheses DE over BC close parentheses squared

increment ABC space fraction numerator ar left parenthesis increment ADE right parenthesis over denominator 81 end fraction equals 4 over 9

ar left parenthesis increment ADE right parenthesis space equals space 36 space cm squared.

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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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