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Triangles

Short Answer Type

241. In Fig. DE || BC and

space AD over DB equals 3 over 5.

If AC = 4.8 cm, find the length of AE.

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242. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ∆ABE ~ ∆CFB.

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243.

In the given fig., $PS over SQ equals PT over TR$ and ∠PST = ∠PRQ. Prove that ∆PQR is an isosceles triangle.

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244.

In the given fig, ∆ABD is a right triangle, right angled at A and AC ⊥ BD. Prove that AB² = BC. BD.

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245.

In the given fig, AB || DC. Find the value of x.

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246.

In the given fig, express x in terms of a, b and c.

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247.

In the given fig, $OA over OC equals OD over OB.$ Prove that ∠A = ∠C and ∠B = ∠D.

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Long Answer Type

248. The diagonal BD of a parallelogram ABCD intersects the segment AE at the point F, where E is any point on the side BC. Prove that DF × EF = FB × FA.

In ∆AFD and ∆BFE, we have
∠1 = ∠2 [Vertically opposite angles]
∠3 = ∠4 [Alternate angles]
In ∆AFD and ∆BFE, we have∠1 = ∠2 [Vertically opposite angles]

So, by AA-criterion of similarity, we have

increment FBE space tilde space increment FDA

rightwards double arrow

space FB over FD equals FE over FA

rightwards double arrow

FB over DF equals EF over FA

rightwards double arrow

DF cross times EF equals FB cross times FA.

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Short Answer Type

249. In a ∠ABC, BD and CE are the altitudes. Prove that ∠ADB and ∠AEC are similar. Is ∆CDB ~ ∆BEC?

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250. The perimeters of two similar triangles ABC and PQR are respectively 36 cm and 24 cm. If PQ = 10 cm, find AB.

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