Locus (2 Viewers)

[nrlwinner]

We haven't studied this at school yet, so could anyone explain this to me? I've got a couple of questions on it.

In a triangle APB, the side AB remains fixed and the vertex P moves in a place, what is the locus of P if angle APB remains the same size?

In a triangle APB, angle APB is a right angle. What is the locus of P in a plane?

 

[jet]

Both are circles. The first has a chord AB, and the second has a diameter AB.
If you had specific points/angles etc. I could give you specifics.
The thing to realise is that they go with the circle geometry theorems:
i- Angles on the circumference standing on the same arc/chord are equal
ii- The angle in a semicircle is a right angle.

 

[nrlwinner]

Thanks but what exactly is "locus?"

 

[jet]

It is a set of points which obey a/a number of condition/s.
For example, the locus of points equidistant from a fixed point is a circle, with radius equal to the distance.
Another would be the locus of points equidistant from TWO points is a straight line.
A parabola is the locus of points equidistant from a line and a point.
And so on.

 

[nrlwinner]

Can you answer one of the questions because I still don't really understand.

 

[jet]

Well... I can't do anything numerical because I have no numbers. I need points and angles and distances.
I can draw diagrams for you. Just give me 5 minutes.

 

[oly1991]

just think of the locus as a compass.

 

[jet]

Okay, so for your first one, I said it was a circle... It is actually a segment of a circle. If you think about it, in order for APB to stay the same angle, the sides will change lengths in a set way. Turns out this makes a segment. If you had some points for A and B, and an angle at P then I could give you an equation for the segment.
This links to the circle theorem 'Two angles standing on the circumference of a circle subtended by the same chord/arc in the same segment are equal'.

This is a special case where theta = 90°. Because of this, it works in the whole circle. You can link this to the circle theorem 'The angle in a semicircle is 90°'.

 

[nrlwinner]

Okay, so for your first one, I said it was a circle... It is actually a segment of a circle. If you think about it, in order for APB to stay the same angle, the sides will change lengths in a set way. Turns out this makes a segment. If you had some points for A and B, and an angle at P then I could give you an equation for the segment.
This links to the circle theorem 'Two angles standing on the circumference of a circle subtended by the same chord/arc in the same segment are equal'.

This is a special case where theta = 90°. Because of this, it works in the whole circle. You can link this to the circle theorem 'The angle in a semicircle is 90°'.

Thanks. Perfectly explained.

 

[Silverste1n]

Very nice explanation, will bookmark for future reference.