# Geometry (1)

• Nov 7th 2009, 07:33 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, 07:58 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 $\angle BAO = x,\, \angle ABO = y$.

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

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

$\Rightarrow x + y = 45^o$

$\Rightarrow \angle AOB = 135^o$ (angle sum of $\triangle AOB$)
• Nov 7th 2009, 07:44 PM
thereddevils
Quote:

Hello thereddevilsSuppose $\angle BAO = x,\, \angle ABO = y$.

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

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

$\Rightarrow x + y = 45^o$

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