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Question

A student constructed a triangle with the known conditions to him being the perimeter of the triangle and both the base angles. The steps of construction he used are as follows:

1. Draw a line segment, say XY equal to AB + BC + AC

2. Make angles LXY equal to B and MYX equal to C

3. Bisect LXY and MXY. Let these bisectors intersect each other at A.

4. Draw perpendicular bisectors PQ of AX and RS of AY

5. Let PQ intersect XY at B and RS intersect XY at C. Join AB and AC.

6. The triangle ABC is thus formed.

How many isosceles triangles are there in this figure?


A

0

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B

1

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C

2

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D

4

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Solution

The correct option is C

2


If we take XBQ and ABQ, we have

XQ = AQ (perpendicular bisector)

BQX = AQB = 90° (perpendicular bisector)

BQ = BQ (common side)

XBQ ABQ (By SAS Congruency)

XB = AB (By CPCT)

Similarly, taking ACR and YCR, we have

AR = YR (perpendicular bisector)

ARC = YRC = 90°(perpendicular bisector)

CR = CR (common side)

ACR YCR (By SAS Congruency)

AC = CY (By CPCT)

Hence the triangles ABX and ACY are isosceles triangles.

.


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