Outline Mathematics
Geometry

Problem 14 from AMC 8 2007

The base of isosceles triangle ABC is 24 and its area is 60. Find the length of one of the congruent sides.

Solution

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The area of a triangle is half the product of a side by the altitude,median,angle,angle bisector,altitude to that side,side,distance,point,plane- A = h×b/2. From here h = 2A/b,2A×b,2A/b,2b/A,(b + A)/2. We are given that the area of the triangle is 60,56,60,48,24 and the base is 24,24,60,48,56. Substituting this into the formula we just found h = 120/24 = 5. Assume AC is the base of the triangle and M is the midpoint of the base. Then ΔABM is right,isosceles,scalene,acute,right,obtuse and we may apply the Pythagorean theorem after observing that, in ΔABM, the legs (i.e., the sides adjacent,opposite,adjacent,equal to the right angle are known: one of them is the altitude h = 5 while the other is half the base, b/2 = 12. The unknown hypotenuse is then √5² + 12² = 13, which answers the problem.

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