CAT Geometry Sectional Test 3 – with Solutions

CAT Geometry Sectional Test 3: Circle Equations, Triangle Centres and 3D Solids

CAT Geometry Sectional Test 3 develops skill in circle equations, triangle centres and three-dimensional mensuration for the CAT Quant section. It includes applied problems such as a bridge arch, a capsule-shaped tank and an ant crawling over a box. Aptitude360 provides a worked solution for every problem.

Areas tested in CAT Geometry Sectional Test 3

• Circle equations: centre and radius from the general form, tangency of a line, lattice points on a circle
• Triangle centres: orthocentre in coordinates, incircle and excircle touch points, altitude and median from the right angle
• Triangle results: Apollonius-type sums of squared lengths, maximum area for two fixed sides, angles in arithmetic progression
• Circle geometry: overlapping equal circles, a circle through two vertices of a square, a bridge arch as a circular arc, rolling coins
• Squares, rectangles and polygons: inscribed squares, the British flag theorem for a point inside a rectangle, star figures in a regular heptagon
• 3D mensuration: spheres related to a cube, capsule tank, square pyramid volume, shortest path on a box surface

Key formulas for circles and solids

• Circle \( x^{2}+y^{2}+2gx+2fy+c=0 \): \( r=\sqrt{g^{2}+f^{2}-c} \)
• Arch of half-span a and height h: \( a^{2}=h(2r-h) \)
• Sphere touching the edges of a cube of side a: \( r=\dfrac{a}{\sqrt{2}} \)
• Sphere through the vertices of a cube: \( r=\dfrac{\sqrt{3}}{2}a \)
• Volume of a sphere: \( V=\dfrac{4}{3}\pi r^{3} \)
• Volume of a pyramid: \( V=\dfrac{1}{3}\times\text{base area}\times h \)

How to revise these CAT geometry topics

• Convert every circle equation to centre and radius form first. A line is tangent when its distance from the centre equals the radius.
• For shortest paths on a box, unfold the faces in each possible way and compare the straight-line distances.
• Learn the incircle and excircle tangent lengths in terms of the semi-perimeter, as they turn long problems into short ones.

Who should take this test

This test is meant for CAT aspirants who are comfortable with basic geometry and want practice on coordinate circles and solid figures.