IMO Practice Test — Quadrilaterals
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
hard
ABCD is a parallelogram where the internal bisectors of two consecutive adjacent angles, Angle A and Angle B, intersect at a point P. What is the exact measure of Angle APB?
45°
60°
90°
120°
Explanation: Since A+B=180°, half of A + half of B = 90°. In Triangle APB, the remaining angle must be 90°.
Question 2 of 6
hard
In an isosceles trapezium ABCD, side AB is parallel to side CD, and non-parallel side AD is equal to side BC. If Angle A = 75°, find the measure of Angle C.
75°
105°
115°
125°
Explanation: In an isosceles trapezium, adjacent base angles are equal (A=B=75°). Since B+C=180°, C = 105°.
Question 3 of 6
hard
In Triangle ABC, AD is the median drawn to the base side BC. A line is drawn through the mid-point of AD, parallel to BC, intersecting side AC at point E. What fraction of AC is the segment AE?
1/4 of AC
1/2 of AC
1/3 of AC
2/3 of AC
Explanation: By the converse of the midpoint theorem on the sub-triangle, E acts as the exact mid-point of AC.
Question 4 of 6
hard
The diagonals of a parallelogram ABCD cross each other at the central origin point O. If a straight line passing through O intersects side AB at P and side CD at Q, then line segment OP is always:
Longer than OQ
Shorter than OQ
Equal to OQ
Double the length of OQ
Explanation: Triangle APO is congruent to Triangle CQO by ASA criteria, which proves OP = OQ via CPCT.
Question 5 of 6
hard
A regular quadrilateral has a perimeter of 40 cm. If its modern coordinate diagonals are identical and intersect perpendicularly, what is the total calculated surface area of this shape?
50 sq cm
100 sq cm
200 sq cm
400 sq cm
Explanation: The clues identify the shape as a square. Side = 40/4 = 10 cm. Area = side squared = 100 sq cm.
Question 6 of 6
hard
If P, Q, R, and S are the mid-points of the sides of a rhombus ABCD, then the inner quadrilateral PQRS formed by connecting these points will always be a:
Square
Rhombus
Rectangle
Trapezium
Explanation: The diagonals of a rhombus are perpendicular, making the mid-point corner boundaries form 90° angles.