# Geometry (1)

• Nov 7th 2009, 06:33 AM
thereddevils
Geometry (1)
In the diagram , X ,Y and Z are the points of contact of tangents BC , CA and AB respectively to a circle with centre O . If angle ACB is a right angle , prove that angle AOB =135 degree .
• Nov 7th 2009, 06:58 AM
Hello thereddevils
Quote:

Originally Posted by thereddevils
In the diagram , X ,Y and Z are the points of contact of tangents BC , CA and AB respectively to a circle with centre O . If angle ACB is a right angle , prove that angle AOB =135 degree .

Suppose \$\displaystyle \angle BAO = x,\, \angle ABO = y\$.

Then
\$\displaystyle \angle OAY = x\$ (congruent \$\displaystyle \triangle\$'s \$\displaystyle OAZ, AOY\$)

\$\displaystyle \angle OBX = y\$ (congruent \$\displaystyle \triangle\$'s \$\displaystyle OBX, OBZ\$)
But
\$\displaystyle 2x + 2y = 90^o\$ (angle sum of \$\displaystyle \triangle ABC\$)

\$\displaystyle \Rightarrow x + y = 45^o\$

\$\displaystyle \Rightarrow \angle AOB = 135^o\$ (angle sum of \$\displaystyle \triangle AOB\$)
• Nov 7th 2009, 06:44 PM
thereddevils
Quote:

Originally Posted by Grandad
Hello thereddevilsSuppose \$\displaystyle \angle BAO = x,\, \angle ABO = y\$.

Then
\$\displaystyle \angle OAY = x\$ (congruent \$\displaystyle \triangle\$'s \$\displaystyle OAZ, AOY\$)

\$\displaystyle \angle OBX = y\$ (congruent \$\displaystyle \triangle\$'s \$\displaystyle OBX, OBZ\$)
But
\$\displaystyle 2x + 2y = 90^o\$ (angle sum of \$\displaystyle \triangle ABC\$)

\$\displaystyle \Rightarrow x + y = 45^o\$

\$\displaystyle \Rightarrow \angle AOB = 135^o\$ (angle sum of \$\displaystyle \triangle AOB\$)