A computer graphics program displays images on a rectangular screen whose coordinate system has the origin at the bottom-left corner. The screen is 800 pixels wide and 600 pixels high. A circular icon of radius 80 pixels is drawn with its centre at the point A (100, 150). Another circular icon of radius 100 pixels is drawn with its centre at the point B (250, 230). Determine:
- (i) whether any part of either circle lies outside the screen.
- (ii) whether the two circles intersect each other.
Step-by-step solution
Idea: A circle reaches exactly one radius to the left, right, above and below its centre, so compare those extreme values with the screen’s edges. Two circles cross when the distance between their centres is less than the sum of the radii (and more than their difference).
(i) whether any part of either circle lies outside the screen.
- Circle A: leftmost x = 100 − 80 = 20, rightmost 100 + 80 = 180; lowest y = 150 − 80 = 70, highest 150 + 80 = 230.½ mark
- Circle B: leftmost x = 250 − 100 = 150, rightmost 250 + 100 = 350; lowest y = 230 − 100 = 130, highest 230 + 100 = 330.½ mark
- The screen covers 0 ≤ x ≤ 800 and 0 ≤ y ≤ 600. All these values are inside those ranges (20 > 0, 70 > 0, 350 < 800, 330 < 600).½ mark
- So no part of either circle lies outside the screen.½ mark
(ii) whether the two circles intersect each other.
- Distance between the centres: AB = √((250 − 100)2 + (230 − 150)2) = √(1502 + 802) = √(22500 + 6400) = √28900 = 170 pixels.1 mark
- Sum of the radii = 80 + 100 = 180. Since 170 < 180, the circles are close enough to overlap. Also 170 > 100 − 80 = 20, so neither circle is inside the other.½ mark
- So yes, the two circles intersect (they cross at two points and overlap a little).½ mark
Check: Along AB, circle A reaches 80 pixels from A and circle B reaches 100 pixels from B; 80 + 100 = 180 is 10 more than AB = 170, so the circles overlap by 10 pixels along that line ✓.
Answer to write in the exam
(i)
Circle A: 100 − 80 = 20 ≤ x ≤ 180 = 100 + 80; 150 − 80 = 70 ≤ y ≤ 230 = 150 + 80
Circle B: 250 − 100 = 150 ≤ x ≤ 350; 230 − 100 = 130 ≤ y ≤ 330
Screen: 0 ≤ x ≤ 800, 0 ≤ y ≤ 600; all values lie within these
∴ No part of either circle lies outside the screen.
(ii)
AB = √((250 − 100)2 + (230 − 150)2) = √(22500 + 6400) = √28900 = 170
r1 + r2 = 80 + 100 = 180; r2 − r1 = 20
20 < AB = 170 < 180
∴ The two circles intersect (at two points).
Common mistakes that cost marks
- Comparing the distance between the centres with only one radius. For crossing, compare it with the sum of the radii.
- Checking only the right and top edges. A circle can also stick out on the left or bottom; check all four.
- Arithmetic: 1502 + 802 = 22500 + 6400 = 28900, and √28900 = 170 (not 230 = 150 + 80).
How this can come in the exam
Two circles have radii 5 and 3, and their centres are 10 units apart. The circles
- do not meet
- touch at one point
- cross at two points
- lie one inside the other
Show answer
(A) do not meet
Sum of radii = 8 < 10, so they are too far apart to meet.
A phone screen is 1080 pixels wide and 1920 pixels high, with the origin at its bottom-left corner. An app places a round button of radius 60 pixels with its centre at P (1050, 200), and another of radius 50 pixels centred at Q (900, 200).
(i) Does the first button fit on the screen? (ii) Do the two buttons overlap? (iii) What is the largest x-coordinate the first button’s centre can have so that it just fits?
Show answer
(i) Rightmost x = 1050 + 60 = 1110 > 1080, so no, 30 pixels are cut off (1½ marks). (ii) PQ = 1050 − 900 = 150 > 60 + 50 = 110, so no overlap (1½ marks). (iii) x + 60 ≤ 1080 ⇒ x ≤ 1020 (1 mark).Try one yourself
On the same 800 × 600 screen, a circle of radius 250 pixels has its centre at (300, 400). Does any part of it lie outside the screen?
Show answer
Top = 400 + 250 = 650 > 600, so yes: 50 pixels are cut off at the top. (Left 50, right 550 and bottom 150 are inside.)
More questions like this
- Plot the points A (2, 1), B (−1, 2), C (−2, −1), and D (1, −2) in the coordinate plane. Is ABCD a square? Can you explain why? What is the area of this square?
- Let us examine the figure to understand the layout of the room. Notice that this only shows the map of the floor. Do you see why the position of the windows cannot be marked on this map?
- The figure shows Reiaan’s room with points OABC marking its corners. The x- and y-axes are marked in the figure. Point O is the origin.
Referring to the figure, answer the following questions: - 1. What are the standard widths for a room door? Look around your home and in school.
2. Are the doors in your school suitable for people in wheelchairs? - So far, we have only considered points on the two coordinate axes. What can you say about the coordinates of points that are not on either axes?