Question 465075: Consider the right triangle ABC with integer sides. If the area of the triangle is 210, and AC=37 then BC=?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
You don't mention which of the vertices of the triangle is the right angle. So let's assume that Point C is the right angle vertex. Given that, the product of AC and BC must be 210. However, 37 is not an even divisor of 210, violating the restriction that the triangle has integer sides. Hence, angle C is NOT the right angle.
Let Angle B be the right angle, hence AC is the hypotenuse, so the relationship:
must hold and further, since the area is 210, the relation:
must also hold and substituting:
Then substituting  and simplifying we get:
which has irrational roots meaning BC is irrational and violates the integer sides restriction.
I'll leave it as an exercise for the student to show that selecting point A as the right angle leads to a similar conclusion. There is no such triangle that fits the given conditions.
John
My calculator said it, I believe it, that settles it
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