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Q30·CSAT · Prelims 2023

Distinct triangles from points on square sides

ReasoningGeometry & CombinatoricsCube & geometry● Medium

Question

ABCD is a square. One point on each of AB and CD; and two distinct points on each of BC and DA are chosen. How many distinct triangles can be drawn using any three points as vertices out of these six points?

Options

a

16

b

18

c

20

Answer
d

24

Explanation

We have a total of 6 chosen points distributed along the perimeter of a square. Calculate the unconstrained number of ways to choose any 3 points from the total pool of 6: Total combinations = 6{3} = 6 × 5 × 4/3 × 2 × 1 = 20.

A triangle is valid unless the 3 selected points happen to lie on the exact same straight line (collinear). Review the points placement along the square sides:

Side AB: 1 point (Cannot form 3 collinear points)
Side CD: 1 point (Cannot form 3 collinear points)
Side BC: 2 points (Cannot form 3 collinear points)
Side DA: 2 points (Cannot form 3 collinear points)

Since no single straight side segment contains 3 or more points, it is impossible to pick a triplet of collinear points. Zero combinations degenerate. The total number of valid triangles remains exactly 20.

The number of triangles formed by N points is N{3} - \Sigma k{3}, where k represents the number of points clustered along individual collinear segments.

Answer: (c).

Question details

Year

2023

Paper

CSAT

Question

Q30

Section

Logical & Analytical Reasoning

Sub-topic

Geometry & Combinatorics

Type

Cube & geometry

Difficulty

Medium

Source hint

Spatial reasoning

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