Quant
Geometry & Mensuration
Every CAT Geometry & Mensuration formula on one page — triangles, circles, coordinate geometry, and the volume and surface area of cubes, cylinders, cones and spheres — each with a worked example.
View
Coordinate Geometry
CAT'26 QUANT CHEATSHEET
Every coordinate geometry formula you need for CAT 2026 — on one page.
Coordinate Geometry on CAT is algebra wearing a graph — every question is really about distances, ratios, and equations of lines, just expressed through x-y points instead of raw numbers. Once you have the distance, section and slope formulas down, most of the topic becomes substitution: build a line’s equation, check a distance, or read a circle’s centre straight off its equation. This sheet lays out every formula you need for the CAT quant section: distance, section and midpoint formulas, every standard form of a line’s equation, parallel and perpendicular conditions, point-to-line and line-to-line distances, triangle area and collinearity, circles, family of lines, and locus and reflection, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Coordinate Geometry: every formula you need
1Distance Formula
The straight-line distance between any two points.
d = √[(x2−x1)2+(y2−y1)2]
Example: (1,2) and (4,6) → √(9+16) = √25 = 5.
CAT Hack: If the answer needs to be an integer, look for a 3-4-5-style right-triangle gap between the coordinates.
2Section Formula
The point that divides a segment in a given ratio.
internal m:n → (mx2+nx1)/(m+n), (my2+ny1)/(m+n)
Example: A(1,1), B(7,7), ratio 1:2 → ((7+2)/3, (7+2)/3) = (3, 3).
Common Mistake: External division flips a sign: use (m−n) in the denominator instead of (m+n).
3Midpoint & Centroid
The balance point of a segment, and of a triangle's three vertices.
Midpoint = (x1+x2)/2, (y1+y2)/2 | Centroid = average of all 3 vertices
Example: triangle (0,0), (6,0), (3,9) → centroid = (9/3, 9/3) = (3, 3).
4Slope of a Line
How steep a line is, from any two points on it.
m = (y2−y1) / (x2−x1)
Example: (2,3) and (5,9) → (9−3)/(5−2) = 6/3 = 2.
CAT Insight: A vertical line has an undefined slope — the denominator becomes zero.
5Equation of a Line: Slope Forms
Build a line's equation from its slope and one known point.
slope-intercept: y=mx+c | point-slope: y−y1=m(x−x1)
Example: slope 2 through (1,3) → y−3=2(x−1) → y = 2x+1.
6Equation of a Line: Two-Point & Intercept Forms
Build a line's equation from two points, or from its axis intercepts.
two-point: (y−y1)/(y2−y1) = (x−x1)/(x2−x1) | intercept: x/a+y/b=1
Example: through (1,1) and (3,5) → slope=2 → y = 2x−1.
7Parallel & Perpendicular Lines
How the slopes of two lines relate depending on their orientation.
parallel: m1 = m2 | perpendicular: m1·m2 = −1
Example: slope 3 and slope −1/3 → product = −1 → perpendicular.
CAT Hack: Given one line's slope m, a perpendicular line's slope is always −1/m — flip and negate.
8Angle Between Two Lines
The acute angle formed where two lines cross.
tanθ = |(m1−m2) / (1+m1m2)|
Example: m1=1, m2=0 → tanθ=|(1−0)/(1+0)|=1 → θ=45°.
9Distance of a Point from a Line
The perpendicular distance from a point to a line Ax+By+C=0.
d = |Ax1+By1+C| / √(A2+B2)
Example: point (2,3), line 3x+4y−6=0 → |6+12−6|/5 = 12/5 = 2.4.
Common Mistake: Don't drop the absolute value — the raw numerator can come out negative before you take it.
10Distance Between Two Parallel Lines
Parallel lines share A and B, so only the constants differ.
d = |C1−C2| / √(A2+B2)
Example: 3x+4y−6=0 and 3x+4y+9=0 → |−6−9|/5 = 15/5 = 3.
11Area of a Triangle (Coordinates)
Compute a triangle's area directly from its three vertices.
Area = ½|x1(y2−y3)+x2(y3−y1)+x3(y1−y2)|
Example: (0,0), (6,0), (0,8) → ½|48| = 24.
CAT Favourite: This one formula handles both triangle area and collinearity checks — a genuine CAT favourite.
12Collinearity of Three Points
Three points lie on one line exactly when their triangle's area is zero.
points collinear ⇔ area of the triangle they form = 0
Example: (1,1), (2,2), (3,3) → area = ½|−1+4−3| = 0 → collinear.
13Equation of a Circle
The standard and general forms of a circle's equation.
(x−h)2+(y−k)2=r2 | general: x2+y2+2gx+2fy+c=0
Example: centre (2,3), radius 5 → (x−2)2+(y−3)2=25.
CAT Insight: From the general form, the centre is (−g, −f) and radius = √(g2+f2−c) — watch the sign flip on g and f.
14Family of Lines
A single equation that represents every line through a fixed intersection point.
L1 + λL2 = 0 passes through the intersection of L1=0 and L2=0
Example: (x+y−3)+λ(2x−y)=0 passes through the intersection of x+y=3 and 2x=y, for any λ.
CAT Hack: Use this to skip solving for the intersection point directly when a question only needs a line through it.
15Locus & Reflection
The path traced by points satisfying a condition, and mirrored points.
reflect about x-axis: (x,−y) | about y-axis: (−x,y) | about y=x: (y,x)
Example: reflect (3,4) about the x-axis → (3, −4).
CAT exam shortcuts, traps & revision
16
CAT Exam Shortcuts
- Distance = √[(x2−x1)2+(y2−y1)2]; midpoint = average of coordinates
- Parallel: m1=m2; perpendicular: m1m2=−1 (flip and negate)
- Point-to-line distance = |Ax+By+C| / √(A²+B²)
- Area of a triangle from coordinates also tests collinearity (area = 0)
- Circle general form: centre = (−g, −f), radius = √(g²+f²−c)
- Family of lines L1+λL2=0 skips solving for the intersection directly
17
Most Common CAT Traps
- Dropping the absolute value in the point-to-line distance formula.
- Using (m−n) instead of (m+n) in the section formula for internal division, or vice versa for external.
- Getting the perpendicular-slope condition backwards (it's a product of −1, not equal slopes).
- Reading the centre straight off the general circle equation as (g, f) instead of (−g, −f).
- Forgetting that a vertical line has an undefined (not zero) slope.
18
30-Second Revision Box
- Distance, midpoint, section and centroid formulas
- Slope m=(y2−y1)/(x2−x1); parallel m1=m2; perpendicular product=−1
- Line forms: slope-intercept, point-slope, two-point, intercept
- Point-to-line and line-to-line distance both use √(A²+B²) in the denominator
- Area of a triangle from coordinates; zero area means collinear
- Circle: centre (−g,−f), radius √(g²+f²−c)
This topic rewards setting up the right formula over solving from scratch — once you recognise a question wants a slope condition, a point-to-line distance, or a family of lines, the algebra is just substitution. Drill this sheet until the slope conditions and the circle’s general form are reflex, then test them on full sets and track progress with the CAT score predictor. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
