    Question

# A student constructed a triangle when the perimeter of the triangle and both the base angles are given. The steps of construction he used are as follows: 1. Draw a line segment, say XY equal to AB + BC + AC. 2. Make ∠LXY equal to ∠B and ∠MYX equal to ∠C. 3. Bisect ∠LXY and ∠MYX. Let these bisectors intersect each other at A. 4. Draw perpendicular bisectors DE of AX and FG of AY. 5. Let BQ intersect XY at B and RC 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 ACYare isosceles triangles.  Suggest Corrections  0      Related Videos   Perimeter and Base angles - proof
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