Question

A square is inscribed in an isosceles right triangle so that squares and the triangle have one common angle show that the vertex of the square and the vertex of the common angle bisects the hypotenuse.

Answer 1

let the square be CMPN
all sides are equal so
CM=MP=PN=CN

ΔABC is isoceles so AC=BC
AN+NC= CM+MB
as CN= CM so.
AN=MB

now consider ΔANP and ΔPMB
AN=MB
∠ANP= ∠PMB= 90
PN=PM
so both triangles are congurent
and AP=PB

Answer 2

Step-by-step explanation:

let the square be CMPN

all sides are equal so

CM=MP=PN=CN

ΔABC is isoceles so AC=BC

AN+NC= CM+MB

as CN= CM so.

AN=MB

now consider ΔANP and ΔPMB

AN=MB

∠ANP= ∠PMB= 90

PN=PM

so both triangles are congurent

and AP=PB.