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:

$\text{1. Draw a line segment, say XY equal to}\text{AB + BC + AC.}$

$\text{2. Make}\angle LXY\text{equal to}\angle B\text{and}\angle MYX\text{equal to}\angle C.$

$\text{3. Bisect}\angle LXY\text{and}\angle MYX.\text{Let these}\text{bisectors intersect each other at A.}$

$\text{4. Draw perpendicular bisectors DE of AX}\text{and FG of AY.}$

$\text{5. Let BQ intersect XY at B and RC}\text{intersect XY at C. Join AB and AC.}$

$\text{6. The triangle ABC is thus formed.}$

$\text{How many isosceles triangles are there}\text{in this figure?}$

• A. 2
• B. 4
• C. 1
• D. 0

Answer 1

The correct option is A 2

$\text{If we take}△XBQ\text{and}△ABQ,\text{we have}$

$XQ=AQ\text{(Perpendicular bisector)}\angle BQX=\angle AQB=90°\text{(Perpendicular bisector)}BQ=BQ\text{(Common side)}$

$△XBQ\cong △ABQ\text{(By SAS Congruency)}$

$\Rightarrow XB=AB\text{(By CPCT)}$

$\text{Similarly, taking}△ACR\text{and}△YCR,\text{we have,}$

$AR=YR\text{(Perpendicular bisector)}\angle ARC=\angle YRC=90°\text{(Perpendicular bisector)}CR=CR\text{(Common side)}$

$△ACR\cong △YCR\text{(By SAS Congruency)}$

$\Rightarrow AC=CY\text{(By CPCT)}$

$\text{Hence, the triangles ABX and ACY}\text{are isosceles triangles.}$