Q7a)i) 2005 hsc paper (1 Viewer)

[obliviousninja]

^^^
I'm having trouble, and I think the answers are wrong.

 

[chris_power_96]

-Find PT, through Pythag, which gives you 2.05
-Then realise that the triangle which subtends alpha is isoceles as it has equal sides (tangent from external pt)
-Then bisect the triangle that holds alpha into two triangles, through taking a line from p to the centre of the circle, which gives equal triangles, thus bisecting alpha
-then using the length PT and tan rule and the fact that one of the bisected triangles is right angled, as tangent to radius subtends right angle, find r

 

[obliviousninja]

-Find PT, through Pythag, which gives you 2.05
-Then realise that the triangle which subtends alpha is isoceles as it has equal sides (tangent from external pt)
-Then bisect the triangle that holds alpha into two triangles, through taking a line from p to the centre of the circle, which gives equal triangles, thus bisecting alpha
-then using the length PT and tan rule and the fact that one of the bisected triangles is right angled, as tangent to radius subtends right angle, find r

Yea, thats pretty much the answers, my concern is just the second step. I dont understand how its isosceles?

 

[chris_power_96]

Yea, thats pretty much the answers, my concern is just the second step. I dont understand how its isosceles?

As tangents from an external point are equal (circle geo), meaning two sides are equal and isoceles triangle

 

[obliviousninja]

As tangents from an external point are equal (circle geo), meaning two sides are equal and isoceles triangle

How do we know they are tangents?

 

[Sy123]

How do we know they are tangents?

For the second step, call the point X below P.
(the one 450 m below it)

Call the points where the tangents intersect the circle M and N.

We know that MX = NX, because tangents from external point are equal.
We know that angle PXM = angle PXN = 90 degrees, since X is directly beneath P.

Therefore, due to Pythagoras theorem

PM^2 = PX^2 + XM^2

PN^2 = PX^2 + XN^2

Since, XM = XN, therefore PM = PN

Therefore the triangle is isosceles.

 

[obliviousninja]

For the second step, call the point X below P.
(the one 450 m below it)

Call the points where the tangents intersect the circle M and N.

We know that MX = NX, because tangents from external point are equal.
We know that angle PXM = angle PXN = 90 degrees, since X is directly beneath P.

Therefore, due to Pythagoras theorem

PM^2 = PX^2 + XM^2

PN^2 = PX^2 + XN^2

Since, XM = XN, therefore PM = PN

Therefore the triangle is isosceles.

How can it be 90, if angle PXT=90

 

[Sy123]

How can it be 90, if angle PXT=90

Why not?

The lines XT and XM and XN lie on the same plane or 'ground level'. If angle PXT is 90 then angle PXM is 90 and PXN is 90

 

[obliviousninja]

My bad, im interpretting as a 2d diagram. ffs, im so stupid.