<p>Consider the five points comprising of the vertices of a square and the intersection point of its diagonals. The number of triangles that can be formed using these points is ___.</p>
Step-by-Step Solution
Key Concept: From 5 points, we can form C(5,3) = 10 total combinations of 3 points, but we must subtract the collinear triplets (points lying on the same line) since they cannot form a triangle.
<p><strong>Step 1:</strong> Label the square vertices as A, B, C, D (in order) and the center as O (intersection of diagonals).</p><p><strong>Step 2:</strong> Total ways to choose 3 points from 5: C(5,3) = 10</p><p><strong>Step 3:</strong> Identify collinear triplets:</p><ul><li>Diagonal AC passes through: A, O, C → collinear triplet {A, O, C}</li><li>Diagonal BD passes through: B, O, D → collinear triplet {B, O, D}</li></ul><p>Total collinear triplets = 2</p><p><strong>Step 4:</strong> Number of valid triangles = Total combinations − Collinear triplets</p><p>= 10 − 2 = 8</p><p>∴ Answer: <strong>8</strong></p>
Correct Answer: 8