Problem 7 · 2024 Math Kangaroo
Medium
Algebra & Patterns
caseworksubstitution
Let a and b be numbers from the set {1, 2, 3, 4, 5, 6}. For each pair (a, b) we draw the line \(y = ax + b\) and look at the triangle it makes with the coordinate axes. For how many pairs (a, b) is that triangle isosceles?
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Answer: D — 6
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Hint 1 of 2
The line y = ax + b cuts the axes at (0, b) and (-b/a, 0); these legs make a right triangle.
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Hint 2 of 2
A right triangle is isosceles exactly when its two legs are equal.
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Approach: set the two axis intercept lengths equal
- The right triangle has legs of length |b| (on the y-axis) and |b/a| (on the x-axis).
- It is isosceles when these are equal: |b| = |b/a|, i.e. |a| = 1.
- Among a in {1,...,6} only a = 1 works, and b can be any of the 6 values, giving 6 pairs.
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