Prove that if XYZ is a triangle with angle X30 Y60 Z90 and W

Prove that, if XYZ is a triangle with angle X=30°, Y=60°, Z=90° and W is the midpoint of the hypotenuse, then the line connecting W to Z divides XYZ into an equilateral triangle and an isosceles triangle.

Solution

Solution:

Use the trigonometry principle

SinX/x = SinY/y = SinZ/z= k

hence , sin30/x = sin60/y = sin90/z= k

hence , x= 1/2k , y= (3)^1/2/2k and z= 1/k

Area of XYZ= 1/2(base )(height)= 1/2 .((3)^1/2/2k)(1/2k) =((3)^1/2 )/8k²

Hence , half area of XYZ= area of iso. triangle= ((3)^1/2 )/16k² and

half area of XYZ= area of equilateral triangle= ((3)^1/2 )/16k² = (((3)^1/2 )a²)/4 on solving this we get a= 1/2k

this gives the length of the side of WZ= a= 1/2k

Hence the sides of the divide triangle are, XW= 1/2k, XZ= ((3)^1/2)/2k , WZ= 1/2k ...............hence Isosceles triangle

and WY= 1/2k , YZ= 1/2k, WZ= 1/2k ........................................hence equilateral triangle