IMO Practice Test — Mid-point Theorem

In a quadrilateral ABCD, the mid-points of all four sides are connected in sequence to form an inner four-sided shape PQRS. This new inner shape PQRS is guaranteed to be a:

In a triangle ABC, AD is the median drawn to the base side BC. A line segment EF is drawn connecting the mid-points of sides AB and AC. If EF intersects the median line AD at a point G, determine the ratio of segment AG to GD.

In a right-angled triangle ABC, where the right angle is located at vertex B, the points P, Q, and R represent the mid-points of sides AB, BC, and CA respectively. If AB = 6 cm and BC = 8 cm, find the absolute length of segment PR.

In a trapezium ABCD, side AB runs parallel to side DC. Points E and F are chosen as the exact mid-points of the diagonals AC and BD respectively. If AB = 10 cm and DC = 6 cm, find the length of segment EF.

Three horizontal lines run parallel to one another. A transversal line crosses them at a 30-degree tilt, creating equal intercepts of length 6 cm. A second transversal crosses them at a right angle (90 degrees). Find the length of the intercepts formed on this second transversal.

Inside a large triangle ABC, a line segment DE is drawn joining the mid-points of AB and AC. A point P is chosen anywhere along the base line BC. If the area of the large triangle ABC is 32 square centimeters, find the area of the smaller triangle PDE.