No questions found
The diagrams show the number of regions inside a circle when 1 radius and 2 radii are drawn. The regions inside the circle are numbered.
(a) (i) Draw radii on the circles below and number the regions.
(ii) Complete the table.
[Table_1]
Number of radii | Number of regions
1 | 1
2 | 2
3 | ext{ }
4 | ext{ }
5 | ext{ }
6 | ext{ }
(b) Write a formula, in terms of $n$, for the number of regions, $R$, when there are $n$ radii.
503
555
540
Diameters
The diagrams show the number of regions inside a circle when 1 diameter and 2 diameters are drawn.
(a) Complete the table for 3, 4 and 5 diameters. You may use the empty circle to help you.
[Table_1: Number of diameters vs Number of regions]
| Number of diameters | Number of regions |
|---------------------|------------------|
| 1 | 2 |
| 2 | 4 |
| 3 | |
| 4 | |
| 5 | |
[3 marks]
(b) Write a formula, in terms of $n$, for the number of regions, $R$, when there are $n$ diameters.
..................................................... [2 marks]
575
558
521
Chords
In this investigation:
• each chord must cut every other chord
• only two chords may intersect at any point.
The diagrams show the number of regions inside a circle when 1 chord, 2 chords and 3 chords are drawn.
(a) Count the number of regions in the circle when 4 chords are drawn.
.................................................. [1]
(b) Complete this table. You may use the empty circle to help you.
[Table_1]
[2]
(c) This is a formula for the number of regions, $R$, when there are $n$ intersecting chords.
$$R = \frac{1}{2}n^2 + bn + 1$$
Find the value of $b$. .................................................. [3]
520
550
596
Tangents
A region can be inside or outside the circle when the lines are tangents.
These two diagrams both show a circle with 2 tangents and the regions numbered.
The maximum number of regions for a circle with 2 tangents is 6.
(a) Give a reason why the first diagram does not have the maximum number of regions with 2 tangents.
.......................................................... ......................................................................................................... ................... [1]
(b) (i) Use this diagram to find the maximum number of regions when there are 3 tangents.
......................................................... [1]
(ii) Draw a fourth tangent on the diagram below to find the maximum number of regions.
............................................................... [2]
(c) Use your answers to part (b) to complete the table.
[Table_1]
(d) This is a formula for the maximum number of regions, $R$, when there are $n$ tangents.
$$R = \frac{1}{2}n^2 + bn + 1$$
Find the value of $b$.
................................................................. [3]
574
596
593
Secants
A secant is a straight line that intersects a circle at two points and extends outside the circle.
In this investigation:
• each secant must cut every other secant
• only 2 secants may intersect at any point
• secants must not intersect on the circumference of the circle.
The diagrams show the number of regions when 1 secant and 2 secants are drawn.
(a) Draw a third secant on the diagram below to find the number of regions when there are 3 secants. Complete the table.
Number of secants | Number of regions
| --- | --- |
| 1 | 4 |
| 2 | 8 |
| 3 | |
| 4 | 19 |
| 5 | 26 |
[2]
(b) This is a formula for the number of regions, $R$, when there are $n$ secants.
$$R = \frac{1}{2}n^2 + bn + c$$
Find the value of $b$ and the value of $c$.
$$b = \text{.....................}$$
$$c = \text{.....................}$$ [5]
596
588
563
There are two circles. The first circle has chords drawn on it. The second circle has tangents drawn on it.
The number of chords on the first circle is the same as the number of tangents on the second circle.
Each circle has the maximum number of regions. One circle has 60 more regions than the other.
(a) Find the number of straight lines on each diagram. ..................................................... [2]
(b) Find the larger number of regions. ..................................................... [2]
595
530
577
