Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Find the distance between (−3, −5) and (9, 0).
Show worked solution
x: 9 − (−3) = 12; y: 0 − (−5) = 5. d2 = 144 + 25 = 169, so d = 13. 17 adds 12 + 5; 169 forgets the square root; √61 uses 9 − 3 = 6, a sign error.
2 Triangle PQR has P(−4, −2), Q(5, 10) and R(5, −2). Find the length PQ, and the perimeter of the triangle.
Show worked solution
PR = 5 − (−4) = 9 and QR = 10 − (−2) = 12. PQ2 = 81 + 144 = 225, so PQ = 15. Perimeter = 9 + 12 + 15 = 36.
3 Find the perimeter of the quadrilateral ABCD with A(−4, 1), B(−1, 5), C(3, 2) and D(0, −2).
Show worked solution
AB2 = 32 + 42 = 25, BC2 = 42 + (−3)2 = 25, CD2 = (−3)2 + (−4)2 = 25 and DA2 = (−4)2 + 32 = 25. Every side is 5, so the perimeter is 20.
Level 2 · Core
4 Which point is 13 units from C(2, −3)?
Show worked solution
(7, 9): 52 + 122 = 25 + 144 = 169 = 132, so (7, 9). (−7, 9): 92 + 122 = 225, a distance of 15. (8, 4): the differences 6 and 7 ADD to 13, but 62 + 72 = 85. (10, −8): 8 and 5 add to 13 too, but 82 + 52 = 89.
5 A(−1, k) and B(5, 3) are 10 units apart. Find both possible values of k.
Show worked solution
36 + (3 − k)2 = 100, so (3 − k)2 = 64 and 3 − k = 8 or −8. So k = −5 or k = 11.
6 The point P(a, 0) on the x-axis is the same distance from A(1, 4) as from B(7, 2). Find a.
Show worked solution
(a − 1)2 + 42 = (a − 7)2 + 22, so a2 − 2a + 1 + 16 = a2 − 14a + 49 + 4. The a2 terms cancel: 12a = 36, so a = 3. Check: PA2 = 4 + 16 = 20 and PB2 = 16 + 4 = 20 ✓.
Level 3 · Challenge
7 Three statements:
1 A(−2, 1), B(2, 4) and C(5, 0) form a right-angled isosceles triangle.
2 For any numbers a and b, the distance from (a, b) to (a + 5, b − 12) is 13.
3 The point (4, −3) is closer to the origin than (−2, 4) is.
Which statements are correct?
1 A(−2, 1), B(2, 4) and C(5, 0) form a right-angled isosceles triangle.
2 For any numbers a and b, the distance from (a, b) to (a + 5, b − 12) is 13.
3 The point (4, −3) is closer to the origin than (−2, 4) is.
Which statements are correct?
Show worked solution
AB2 = 42 + 32 = 25, BC2 = 32 + (−4)2 = 25 and AC2 = 72 + (−1)2 = 50: two equal sides, and 25 + 25 = 50 (1 ✓). The differences are 5 and −12, and 25 + 144 = 169 = 132 whatever a and b are (2 ✓). (4, −3) is √25 = 5 from the origin, but (−2, 4) is √20, which is less than 5, so (4, −3) is farther away (3 ✗). 1 and 2.
8 A is (−3, −1) and B is (5, −1). The point C(1, k), with k > 0, makes AC = BC = 5. Find k, and the perimeter of triangle ABC.
Show worked solution
16 + (k + 1)2 = 25, so (k + 1)2 = 9 and k + 1 = 3 or −3, giving k = 2 or −4. As k > 0, k = 2. AB = 5 − (−3) = 8, so the perimeter is 5 + 5 + 8 = 18.
9 ABCD is a square. A is (−3, 2) and the next corner, B, is (2, 4). What is the area of the square?
Show worked solution
AB2 = (2 − (−3))2 + (4 − 2)2 = 25 + 4 = 29. The area of a square is the side squared, so the area is 29. No square root is needed.