IMO Practice Test — Triangles
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
hard
In Triangle ABC, the bisectors of interior Angle B and exterior Angle C intersect at point P. If Angle A = 50°, find the measure of Angle BPC.
25°
50°
100°
40°
Explanation: By using exterior angle theorems, it can be proven that Angle BPC is always half of Angle A.
Question 2 of 6
hard
Point O is an interior point of a Triangle ABC. Which of the following inequality relations is always true for any triangle?
OA + OB + OC < AB + BC + CA
OA + OB + OC > AB + BC + CA
2(OA + OB + OC) > AB + BC + CA
OA + OB + OC = AB + BC + CA
Explanation: Summing the inequalities of triangles OAB, OBC, and OCA gives 2(OA+OB+OC) > AB+BC+CA.
Question 3 of 6
hard
In a triangle, the lengths of two sides are 6 cm and 13 cm. If the third side length is an integer '$x$', find the number of possible integer values that '$x$' can take.
11
12
13
10
Explanation: $13 - 6 < x < 13 + 6$, so $7 < x < 19$. The integers are 8 to 18, which gives 11 values.
Question 4 of 6
hard
In Triangle ABC, AD is the median drawn to side BC. Which of the following structural statements is always true?
AB + AC = 2AD
AB + AC > 2AD
AB + AC < 2AD
AB + AC = AD
Explanation: Extend AD to E such that AD=DE. Triangle ABD≅ECD implies AC+EC > AE, so AB+AC > 2AD.
Question 5 of 6
hard
In right-angled Triangle ABC (with Angle B = 90°), P and Q are points on sides AB and BC respectively. Choose the correct relation regarding the hypotenuse splits:
$AQ^2 + CP^2 = AC^2 + PQ^2$
$AQ + CP = AC + PQ$
$AQ^2 + CP^2 = 2AC^2$
$AQ \times CP = AC \times PQ$
Explanation: Apply the Pythagoras theorem to triangles ABQ, CBP, ABC, and PBQ and add the equations.
Question 6 of 6
hard
In Triangle ABC, Angle A is twice Angle B ($A = 2B$). A point D lies on side AC such that BD bisects Angle B. If CD = 4 cm and AD = 3 cm, find the length of side AB.
5 cm
6 cm
7 cm
8 cm
Explanation: Triangle ABD is isosceles ($AD=BD=3$). Similarity or angle-bisector properties reveal $AB = CD + AD = 4 + 3 = 7$ cm.