Sequences & Series
Summation
Grade 11
Question:
<p>The number of points, having both co-ordinates as integers that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is</p>
<p>861</p>
<p>820</p>
<p>780</p>
<p>901</p>
Step-by-Step Solution
Key Concept: Use Pick's theorem: A = I + B/2 - 1, where I is interior lattice points and B is boundary lattice points. Calculate boundary points on all three sides, then solve for I.
<p><strong>Step 1:</strong> Identify the triangle vertices: A(0,0), B(0,41), C(41,0). This is a right triangle with legs of length 41.</p><p><strong>Step 2:</strong> Calculate the area: A = ½ × 41 × 41 = 840.5</p><p><strong>Step 3:</strong> Count boundary lattice points (B):</p><ul><li>Side AB (x=0, y from 0 to 41): 42 points</li><li>Side AC (y=0, x from 0 to 41): 42 points</li><li>Side BC (x+y=41, x from 0 to 41): 42 points</li><li>Subtract vertices counted thrice: 42 + 42 + 42 - 3 = 123 points</li></ul><p><strong>Step 4:</strong> Apply Pick's theorem: A = I + B/2 - 1</p><p>840.5 = I + 123/2 - 1</p><p>840.5 = I + 61.5 - 1</p><p>840.5 = I + 60.5</p><p>I = 780</p><p>∴ Answer: C (780)</p>
Correct Answer: C