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Triangles

Long Answer Type

311.

In the given Fig, M is the mid-point of side CD of a parallelogram ABCD. The line BM is drawn intersecting AC at L and AD produced at E. Prove that EL = 2 BL.

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Short Answer Type

312.

In the given Fig., ∆ABC is right angled at C and DE ⊥ AB. Prove that ∆ABC ~ ∆ADE, and hence find the lengths of AE and DE.

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Long Answer Type

313. Prove that the ratio of the areas of two similar triangle is equal to the square of the ratio of their corresponding:
(i) altitudes
(ii) angle bisector segments.

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314. In ∆ABC, XY is parallel to BC and it divides ∆ABC into two parts of equal in area. Prove that

space space space BX over AB equals fraction numerator square root of 2 minus 1 over denominator 2 end fraction.

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315.

In the given fig., DE || BC and AD : DB = 5:4. Find $space fraction numerator ar left parenthesis increment DEF right parenthesis over denominator ar left parenthesis increment CFB right parenthesis end fraction$ .

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316.

In the given Fig., PQ || BC and AP: PB = 1:2. Find $space space space fraction numerator ar left parenthesis increment APQ right parenthesis over denominator ar left parenthesis increment ABC right parenthesis end fraction.$

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317. Two isosceles triangles have equal vertical angles and their areas are in the ratio 16: 25. Find the ratio of their corresponding heights.

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318.
In the given Fig, $XP over PY equals XQ over QZ equals 3 comma$ if the area of XYZ is 32 cm², then find the area of the quadrilateral PYZQ.

We have,

[Using converse of Basic proportionality theorem]
Now, in ∆XPQ and ∆XYZ,
∴ ∠XPQ = ∠XYZ [Corresponding angles]
∠X = ∠Y [Common]
Therefore, by using AA similar condition, we have ∆XPQ ~ ∆XYZ

[∵ The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides]

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319. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

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320. Prove that in any triangle, the sum of the squares of any two sides is equal to twice the square of half of third side together with twice the square of the median which bisects the third side.