Plane geo

anpanman SG

PQ is a chord of the circle centre O. The tangents at P and Q to the circle meet at X and OX intersects PQ at N. Prove that

1. Triangle XPO is similar to triangle XNP

2. ON . OX = OP^2

Tried to prove part 1 using alt. seg. or intersecting chords theorem but realized that both theories can't be applied. Can someone help me out with this?

Thanks!

wee_ws SG

Hi,

1. angle PXO = angle NXP

2. angle XPO = angle XNP = 90 degrees

so by AA, triangle XPO is similar to triangle XNP.

Thanks!

Cheers,
Wen Shih

TrueHeart SG

PQ is a chord of the circle centre O. The tangents at P and Q to the circle meet at X and OX intersects PQ at N. Prove that

1. Triangle XPO is similar to triangle XNP

2. ON . OX = OP^2

Tried to prove part 1 using alt. seg. or intersecting chords theorem but realized that both theories can't be applied. Can someone help me out with this?

Thanks!

Since Mr Wee has kindly posted up the first part, I will elaborate on the second part.

Working from the right side, OP^2 = PN^2 + ON^2 = ON( PN^2/ON + ON ) = ON.OX

From the 1st part, we know that PN/ON = XN/PN, thus ( PN^2/ON + ON ) = OX

wee_ws SG

Hi anpanman,

In case you get stuck in the first part during the exam, just use the result anyway to do the second part. You'll still score a few marks instead of a zero if you give up the entire question.

Thanks and jiayou!

Cheers,
Wen Shih

anpanman SG

Hi,

1. angle PXO = angle NXP

2. angle XPO = angle XNP = 90 degrees

so by AA, triangle XPO is similar to triangle XNP.

Thanks!

Cheers,
Wen Shih

How do you know angle XNP = 90 deg? Thanks!

wee_ws SG

Hi,

Triangle POQ is isosceles. Let angle POQ be theta. Then you should be able to see why. Thanks!

Cheers,
Wen Shih

TrueHeart SG

How do you know angle XNP = 90 deg? Thanks!

Sorry to have extra posted when Mr Wee has already done so.