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Properties of Isosceles Triangle

Through any p...

Question

Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. Then the triangle so formed are:

Scalene

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Equilateral

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Isosceles

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Isoscelesright angled

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Solution

The correct option is C Isosceles
Consider $∠ X Y Z$ , $O Y$ bisects $∠ X Y Z$ , $O P ∥ Y Z$ and $O B ∥ Y Z$
$O Y$ bisects $∠ X Y Z$
Hence, $∠ P Y O = ∠ O Y Z$ (1)
Now, $P O ∥ Y Z$
Hence, $∠ O Y Z = ∠ P O Y$ (Alternate angles) (2)
Thus, from (1) and (2)
$∠ P Y O = ∠ P O Y$
or $△ P O Y$ is an Isosceles triangle
Similarly, $△ O B Z$ is an Isosceles triangle.

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Similar questions

Q. Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. The triangle so formed is an isosceles triangle.If true enter 1 else 0.

Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. Prove that the triangle so formed is isosceles.

Q. Line

l

is the bisector of an angle

∠ A

and

B

is any point on

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B P

and

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are perpendiculars from

B

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(i)

△ A B P ≅ △ A Q B

(ii)

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is equidistant from the arms of

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l

is the bisector of an angle

∠ A

and

B

is any point on

l . B P

and

B Q

are perpendiculars from

B

to the arms of

∠ A

Show that

(i) △ A P B ≅ △ A Q B

(i i) B P = B Q

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Q. In the given figure, line l is the bisector of an angle ∠A and B is any point on l. If BP and BQ are perpendiculars from B to the arms of ∠A, show that
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(ii) BP = BQ, i.e., B is equidistant from the arms of ∠A.

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Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. Then the triangle so formed are:

The correct option is C IsoscelesConsider ∠XYZ, OY bisects ∠XYZ, OP∥YZ and OB∥YZOY bisects ∠XYZHence, ∠PYO=∠OYZ (1)Now, PO∥YZHence, ∠OYZ=∠POY (Alternate angles) (2)Thus, from (1) and (2)∠PYO=∠POYor △POY is an Isosceles triangleSimilarly, △OBZ is an Isosceles triangle.