Help with these two geometry!

xxravenxx · #1 $Old$ October 20th, 2009, 12:37 AM

Help with these two hard problems:

Problem 12
Let M be a point inside the angle A. From point M perpendiculars MP and MQ are dropped to the angle sides. From point A, perpendicular AK is dropped to segment PQ. Prove that angle PAK = angle MAQ.
Problem 13
Let ABC be a triangle and angle B = 60. Angle bisectors AD and CE meet at the point O. Prove that OD = OE.

Grandad · #2 $Old$ October 20th, 2009, 10:16 AM

Hello xxravenxx

Quote:

Originally Posted by xxravenxx $View Post$

... Problem 12
Let M be a point inside the angle A. From point M perpendiculars MP and MQ are dropped to the angle sides. From point A, perpendicular AK is dropped to segment PQ. Prove that angle PAK = angle MAQ.

$\angle APM = \angle AQM = 90^o$ (given)

$\Rightarrow APMQ$ is a cyclic quadrilateral (opp angles supplementary)

$\Rightarrow \angle APQ = \angle AMQ$ (angles in same segment)

So in $\triangle$ 's $APK, AMQ:$

$\angle APK = \angle AMQ$ (proven)

$\angle AKP = \angle AQM$ ( $=90^o$ , given)

$\Rightarrow \angle PAK = \angle MAQ$ (angle sum of triangle)

Grandad

xxravenxx · #3 $Old$ October 21st, 2009, 12:47 AM

Sorry but can you draw me a diagram of what this looks like? I'm a bit clueless.

Grandad · #4 $Old$ October 21st, 2009, 03:13 AM

Hello xxravenxx

Quote:

Originally Posted by xxravenxx $View Post$

Sorry but can you draw me a diagram of what this looks like? I'm a bit clueless.

See attachment.

... and here's the answer for number 13:

Quote:

Originally Posted by xxravenxx $View Post$

...Problem 13
Let ABC be a triangle and angle B = 60. Angle bisectors AD and CE meet at the point O. Prove that OD = OE.

(I hope you can draw a diagram for this.)

Let $\angle BAD = \angle DAC = x^o$ and $\angle ACE = \angle ECB = y^o$

Then, $2x+2y+ 60=180$ (angle sum of $\triangle ABC$ )

$\Rightarrow x+y=60$

Now $\angle DOC$ is an exterior angle of $\triangle OAC$ = sum of interior opposite angles $= x+y = 60^o$

$\Rightarrow BEOD$ is a cyclic quadrilateral (exterior angle $\angle DOC$ = interior opposite angle $\angle B$ )

Now the angle bisectors of a triangle are concurrent $\Rightarrow BO$ is the bisector of $\angle ABC$

$\Rightarrow \angle DBO = \angle EBO$

$\Rightarrow \angle DEO = \angle EDO$ (angles in same segment)

$\Rightarrow OE = OD$ (isosceles $\triangle OED$ )

Grandad