CAT Geometry Sectional Test 5 – with Solutions

CAT Geometry Sectional Test 5: Plane, Coordinate and Solid Geometry Practice

CAT Geometry Sectional Test 5 is a geometry and mensuration practice set for CAT that blends plane geometry, coordinate geometry, heights and distances and solid figures. It includes polygon-based counting, loci and inscribed solids alongside triangle and circle problems. Aptitude360 gives a detailed solution for every question.

What this CAT geometry set tests

• Triangles: sides from angle ratios, distance between circumcentre and incentre, integer-sided triangles with a given angle, isosceles angle chasing
• Quadrilaterals: areas around a point inside a parallelogram, isosceles trapezium with given diagonals, triangle areas inside a rectangle
• Circles: concentric circles and chords, arcs drawn inside a square, circles touching an inscribed circle, intersecting chords and secants
• Coordinate geometry: circle through given points, chord cut on an axis, parallelogram bounded by four lines, locus of a moving point
• Solids and mensuration: rolling cone, cylinder cut along its axis, cone inscribed in a sphere, largest equilateral triangle inside a square
• Other areas: acute triangles from a regular polygon's vertices, pendulum swing angle, angles of elevation to a statue and a pole

Important formulas and rules for this set

• Euler's relation: \( OI^{2}=R^{2}-2Rr \)
• Sine rule: \( \dfrac{a}{\sin A}=2R \)
• Arc length (θ in radians): \( l=r\theta \)
• Curved surface of a cone: \( \pi rl \)
• Parallelogram from lines: \( \dfrac{|(c_{1}-c_{2})(d_{1}-d_{2})|}{|a_{1}b_{2}-a_{2}b_{1}|} \)

How to revise these geometry topics

• For triangles from polygon vertices, remember that a triangle inscribed in a circle is acute only when the centre lies strictly inside it.
• With a point inside a parallelogram, the two triangles on opposite sides together make half the parallelogram.
• A point whose distances from two fixed points stay in a constant ratio other than one traces a circle; find its centre and radius first.
• In heights and distances, sketch each right triangle separately and keep eye level in mind.

Who will gain from this practice test

CAT and OMET aspirants who are comfortable with geometry basics and now want varied, exam-level problems that link algebra, trigonometry and mensuration.