Unit 17: Power of a Point
Lines cutting through circles.
Unit 17 of 28 in Problem-solving maths for kids. Its 7 lessons are Two Chords Crossing, Cutting From Outside, Chords and the Centre, Two Tangents From One Point, Inside, Outside, and On, Circles Meeting Circles and One Theorem All Along — below is everything each one explains, and a question or two from it to try.
The answers are marked in the app by value — 1/2, 0.5 and 2/4 all count — and every one is re-derived by sympy before it ships.
This unit opens with a fortnight’s trial of everything — no card needed — or with a family plan, bought in the iPhone app. The first two units of every track are free for ever. Try it in the app.
❌ Two Chords Crossing
A surprising product
A chord is a straight line joining two points on a circle. Draw two chords so they cross inside.
Each chord is cut into two pieces. Multiply the two pieces of one chord, then the two pieces of the other.
You get the same answer. Every time, wherever the chords are.
Why it works
Join the ends up and you get two triangles. Angles on the same arc are equal — that is the circle rule from unit 14 — so the two triangles are similar.
Similar triangles have matching sides in the same ratio, and rearranging that ratio gives exactly a × b = c × d.
No new magic. Angle chasing plus similar triangles.
Try it yourself
Two chords cross. One is cut into 3 and 8. The other is cut into 4 and what?
Two chords cross. One is cut into 5 and 12. The other is cut into 10 and what?
Answer them in the app
🎣 Cutting From Outside
The same rule, from further away
Now stand outside the circle and send two straight lines through it. Each one enters at one point and leaves at another.
For each line, multiply the distance to the near point by the distance to the far point.
Both lines give the same answer again. It is the same theorem — the crossing point simply moved outside.
A tangent touches just once
A tangent touches the circle at exactly one point and never goes inside.
Think of it as a line whose two crossing points have slid together into one. So the near distance and the far distance are the *same* — and the product becomes that length squared.
tangent² = near × far for any other line from the same point.
Try it yourself
From a point outside, one line reaches the circle at 4 and leaves at 9. Another reaches at 3. Where does it leave?
From a point outside, one line reaches at 5 and leaves at 20. Another reaches at 10. Where does it leave?
Answer them in the app
🎯 Chords and the Centre
The centre always sits on the perpendicular bisector
Take any chord. Find its middle, and draw a line at right angles through that point.
That line passes through the centre of the circle. Always.
The reason is symmetry: both ends of the chord are the same distance from the centre, so the centre must lie on the line of points equidistant from them.
Equal chords sit equally far out
Two chords the same length are the same distance from the centre — and the other way round too.
So a chord close to the centre is long, and one near the edge is short. The diameter, passing right through, is the longest of all.
That gives a quick sanity check: if your answer says a chord 1 cm from the centre is shorter than one 5 cm out, something has gone wrong.
Try it yourself
A chord is 16 long. How far is its midpoint from each end?
That chord is in a circle of radius 10. How far is the chord from the centre?
Answer them in the app
✌️ Two Tangents From One Point
They are always the same length
Stand outside a circle and draw both tangents. They touch at two different points — and the two lengths are equal.
Why: each makes a right angle with its radius, both radii are the same, and the line to the centre is shared. Two right-angled triangles with matching sides.
So the picture is symmetric about that line to the centre, and every tangent question inherits that symmetry.
The tangent length is the power, square-rooted
Here is the link back to lesson 2. The power of an outside point is near × far for any line through the circle.
For the tangent, near and far are the same length t, so the power is t^2.
And from the centre, the power is also d^2 - r^2 by Pythagoras. Two ways to the same number, which is often the quickest route through a problem.
Try it yourself
One tangent from a point is 9 long. How long is the other tangent from the same point?
A point is 15 from the centre of a circle of radius 9. How long is each tangent?
Answer them in the app
🎪 Inside, Outside, and On
Where the point sits changes the sign
The power of a point has a natural meaning: d^2 - r^2, where d is its distance from the centre.
Outside the circle: d > r, so the power is positive — and it equals the tangent squared.
On the circle: d = r, so the power is exactly zero.
Inside: d < r, so the power is negative, which is why the two chord pieces are measured on opposite sides.
One formula covers all three positions.
A chord through an inside point
For a point inside, take the chord through it that passes through the centre — the diameter.
If the point is 3 from the centre of a circle of radius 7, that diameter is cut into 7 - 3 = 4 and 7 + 3 = 10.
Their product is 40 — the size of the power, ignoring the minus. And every other chord through that point gives 40 as well.
Try it yourself
A point is 13 from the centre of a circle of radius 5. What is its power?
What is the power of a point sitting exactly on the circle?
Answer them in the app
🎠 Circles Meeting Circles
Touching from outside
Two circles that touch at exactly one point, each outside the other, have their centres r1 + r2 apart.
So circles of radius 3 and 5 that touch have centres 8 apart, and the touching point sits on the line joining the centres.
That line is the key to every two-circle problem: draw it first.
Touching from inside
A small circle can also touch a big one from the inside. Then the centres are R - r apart.
A circle of radius 3 inside one of radius 10, touching, has centres 7 apart.
If the small circle sits exactly in the middle, the centres coincide and it never touches at all. So "touching" and "inside" are different things.
Try it yourself
Two circles of radius 3 and 5 touch on the outside. How far apart are their centres?
Two circles of radius 7 and 4 touch on the outside. How far apart are their centres?
Answer them in the app
🔗 One Theorem All Along
Three rules, one idea
Chords crossing inside, lines cutting from outside, a tangent touching — all three say the same thing.
For any point and any circle, take a line through the point and multiply the two distances to the circle. You always get the same number, whichever line you pick. That number is the *power* of the point.
Inside the circle, outside it, or touching. One theorem.
Try it yourself
What stays the same for every line drawn through one fixed point?
- The two distances to the circle multiplied together
- The two distances added together
- The length of the line
- The angle it makes
A tangent from P is 15 long. A line from P reaches the circle at 9. Where does it leave?
Answer them in the app