IMO Practice Test — Triangles
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
medium
In Triangle ABC, a line DE is drawn parallel to BC cutting AB at D and AC at E. If Area(ADE) : Area(DECB) = 1 : 3, find the ratio AD : AB.
1:2
1:3
1:4
1:sqrt(3)
Explanation: Total Area(ABC) = 1 + 3 = 4. Area ratio ADE/ABC = 1/4. Side ratio = sqrt(1/4) = 1/2.
Question 2 of 6
medium
An equilateral triangle is described on one side of a square, and another identical equilateral triangle is described on one of its diagonals. Find the ratio of the area of the first triangle to the second triangle.
1:2
1:3
1:4
sqrt(2):1
Explanation: If square side is s, diagonal is s*sqrt(2). Ratio of areas = (s / s*sqrt(2)) squared = 1/2.
Question 3 of 6
medium
In a right-angled triangle ABC, right-angled at B, an altitude BD is dropped perpendicular to the hypotenuse AC. If AD = 4 cm and CD = 9 cm, find the exact length of BD.
5 cm
6 cm
6.5 cm
7.2 cm
Explanation: By similarity of split right triangles, BD^2 = AD * CD. BD = sqrt(4 * 9) = 6 cm.
Question 4 of 6
medium
An airplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another airplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will the two planes be after 1.5 hours?
3300 km
300 * sqrt(61) km
2200 km
150 * sqrt(61) km
Explanation: Distance N = 1500, Distance W = 1800. Separation = sqrt(1500^2 + 1800^2) = 300 * sqrt(5^2 + 6^2) = 300*sqrt(61).
Question 5 of 6
medium
In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3 BC. Find the mathematical relationship between AD squared and AB squared.
7 AD^2 = 9 AB^2
9 AD^2 = 7 AB^2
3 AD^2 = 4 AB^2
4 AD^2 = 3 AB^2
Explanation: Dropping an altitude and using Pythagoras shows AD^2 = 7/9 AB^2, so 9 AD^2 = 7 AB^2.
Question 6 of 6
medium
ABC is an isosceles triangle right-angled at C. Prove that a square built on its hypotenuse AB has a size relation to a square built on side AC. What is this correct relation?
AB^2 = AC^2
AB^2 = 3 AC^2
AB^2 = 2 AC^2
AB^2 = 4 AC^2
Explanation: Since AC = BC, Pythagoras gives AB^2 = AC^2 + BC^2 = AC^2 + AC^2 = 2 AC^2.