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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.

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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?

In ∆ABD and ∆AEC, we have
∠ADB = ∠AEC
[Each equal to 90°]
∠BAD = ∠EAC [Common]
In ∆ABD and ∆AEC, we have∠ADB = ∠AEC[Each equal to 90°]∠BA
So, by AA-criterion of similarity, we have
∆BDA ~ ∆CEA or ∆ADB ~ ∆AEC
Clearly, ∆CDB is not similar to ∆BEC, because they are not equiangular.

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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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