Geometry and Arithmetic account for a major chunk of the Quant section for the CAT Exam. Here are a few interesting true/false questions from Geometry. Happy Solving!
Check out this wonderful article which talks about preparing for Geometry!
CAT Quants | True/False Questions from 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.
Try solving these! For answers and explanations, scroll down AFTER your attempt!
CAT Quants | Solutions to True/False Questions from Geometry
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.
A Final Note:
The CAT 2020 Pattern has changed, leaving many aspirants confused and worried about the number of questions. You might want to check out this article about preparing for these changes in the CAT 2020 exam pattern.
Stay Safe, Best Wishes for CAT!
Rajesh Balasubramanian takes the CAT every year and is a 4-time CAT 100 percentiler. He likes few things more than teaching Math and insists to this day that he is a better teacher than exam-taker.