CAT Prep Tips: Series 1 - Geometry

State whether the following statements are true or false

1. A parallelogram that circumscribes a circle has to be a square
2. A trapezium inscribed in a circle has to be an isosceles trapezium
3. Orthocenter of a triangle can lie outside the triangle
4. Triangle with sides a, b and c has the relationship a^2 + b^2 > c^2, the triangle has to be acute-angled.
5. Diagonals of a parallelogram are angle bisectors of the angles of a parallelogram.

Scroll down for answers and explanation

1. A parallelogram that circumscribes a circle has to be a square: FALSE
In a parallelogram, opposite sides are equal. In a quadrilateral, the sums of pairs of opposite sides are equal. So, a parallelogram that circumscribes a circle should have all 4 of its sides equal. Or, it should be a Rhombus; it need not be a square.

2. A trapezium inscribed in a circle has to be an isosceles trapezium: TRUE
An isosceles trapezium is a symmetric diagram. The two base angles should be equal and the two top angles should be equal. So, a trapezeium where the base angles were equal would be an isosceles trapezium.
In any cyclic quadrilateral, opposite angles would be supplementary. In a trapezium, co-interior angles between the parallel lines would be supplementary. So, if we took a trapezium ABCD with AB parallel to CD inscribed in a circle. Angle A and Angle D would be supplementary (co-interior angles). And Angle A and Angle C would be supplementary (opposite angles of a cyclic quadrilateral). Or angle B would be equal to angle C. Ergo, isosceles trapezium.

3. Orthocenter of a triangle can lie outside the triangle: TRUE
For any obtuse-angled triangle, two of the altitudes would lie outside the triangle, and would intersect at a point outside the triangle. So, the orthocenter can lie outside the triangle.

4. Triangle with sides a, b and c has the relationship a^2 + b^2 > c^2, the triangle has to be acute-angled: FALSE
Let us take triangle with sides 2, 3 and 4. 4^2 + 3^2 > 2^2. But  as 2^2 + 3^3 < 4^2, the triangle is obtuse-angled. Is a^2 + b^2 > c^2, we can say angle C is acute-angled. We cannot say all three angles are acute-angled. One can use cosine rule also for having a go at this question (though it should be considered inelegant)

5. Diagonals of a parallelogram are angle bisectors of the angles of a parallelogram: FALSE
Diagonals of a parallelogram bisect each other. They need not bisect the angles of the parallelogram. Imagine this, if we took a rectangle and studied its diagonals. if the diagonals bisected each other, the angle between diagonal and a side would be 45 degrees. Or, we would end up having a square. So, any rectangle that was not a square would have diagonals that were not angle bisectors. So, diagonals of a parallelogram NEED NOT be angle bisectors of the angles of a parallelogram.