IMO Practice Test — Coordinate Geometry
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
Three vertices of a rectangle are (2, 3), (6, 3), and (2, 7). Find the fourth vertex.
(6, 7)
(7, 6)
(3, 6)
(7, 2)
Explanation: Opposite corners share coordinates: (6,7) completes rectangle
Question 2 of 6
Find the area of triangle with vertices (0,0), (4,0), and (0,3).
6 sq units
12 sq units
10 sq units
8 sq units
Explanation: Area = ½ × base × height = ½ × 4 × 3 = 6
Question 3 of 6
Points \(A(1, 3)\), \(B(5, k)\), and \(C(1, 7)\) form an isosceles triangle with \(AB = BC\). Find \(k\).
3
4
5
6
Explanation: AB²=(5-1)²+(k-3)²=16+(k-3)²; BC²=(1-5)²+(7-k)²=16+(7-k)². AB=BC: (k-3)²=(7-k)² ⇒ k-3=7-k ⇒ 2k=10 ⇒ k=5.
Question 4 of 6
Find the point on the X-axis equidistant from \(A(2, 3)\) and \(B(6, 1)\).
(3, 0)
(4, 0)
(5, 0)
(6, 0)
Explanation: Let P(x,0). PA²=PB²: (x-2)²+9=(x-6)²+1 ⇒ 8x=24 ⇒ x=3. Answer: (3,0).
Question 5 of 6
If points (a,0), (0,b), and (1,1) are collinear, find \(\frac{1}{a} + \frac{1}{b}\).
0
1
-1
2
Explanation: Collinear means slope between (a,0) and (0,b) equals slope between (0,b) and (1,1): (b-0)/(0-a) = (1-b)/(1-0); -b/a = 1-b; Multiply: -b = a(1-b); -b = a - ab; ab - b = a; b(a-1) = a; 1/a + 1/b = (a+b)/ab = ? From b(a-1)=a → b = a/(a-1); Then 1/a + (a-1)/a = 1/a + 1 - 1/a = 1
Question 6 of 6
Find the area of quadrilateral with vertices (0,0), (4,0), (3,3), and (0,3).
10.5 sq units
12 sq units
13.5 sq units
15 sq units
Explanation: This is a trapezium: parallel sides of lengths 4 and 3, height = 3; Area = ½ × (4+3) × 3 = ½ × 7 × 3 = 10.5