# Geometry question?

Keegan is determining whether the triangle with vertices L(−1, 3), M(5, 5) and N(7, −1) is a right triangle. Keegan finds the slopes as shown and concludes that the triangle is not a right triangle because the product is not −1. What is his error and what should he do to correct it?

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• The error might be working with slopes in the first place.

LN is clearly longer than the other two sides, so if a circle with LN as a diameter also passes though M then △LMN has a right angle at M. This is true if (Mx - Lx)(Mx - Nx) + (My - Ly)(My - Ny) = 0

Here (5 - -1)(5 - 7) + (5 - 3)(5 - -1) = (6)(-2) + (2)(6) = 0, so △LMN has a right angle at M

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• If you say "finds the slopes as shown" and then don't show them, it very difficult if not impossible to give a definitive answer to your question.

Slope LM = 2/6 = 1/3

Slope MN = -6/2 = -3

Slope LN = -4/8 = -1/2

LM and MN are legs of a right angle.

I have no way of knowing what Keegan's error is. He may have calculated the slopes wrong or he may not have looked at all the angles in the triangle.

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• Tough to say without seeing what he calculated.  For example, he may have tried sides LN and LM, which are not the two perpendicular sides (LM ans MN are).

Or, he may have calculated slope wrong.

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