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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

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B

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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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