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?
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)
Hence, the triangles ABX and ACYare isosceles triangles.