Sequences & Series
Number Theory in Geometry
Grade 11
Question:
<p>Let a and b be the lengths of the legs of a right triangle with the following properties: (a) All 3 sides of the triangle are integers. (b) The perimeter of the triangle is numerically equal to area of the triangle, it is given that \(a < b\). The number of ordered pairs \((a, b)\) will be:</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) 3</p>
<p>(d) 4</p>
Step-by-Step Solution
Key Concept: Set up equation: Perimeter = Area, i.e., \(a + b + c = \frac{1}{2}ab\) where \(c = \sqrt{a^2 + b^2}\) is an integer (Pythagorean triple), and solve for integer pairs.
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Correct Answer: Unknown