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Question

In right ABC with legs 5 and 12, arcs of circles are drawn, one with center A and radius 12, the other with center B and radius 5. They intersect the hypotenuse in M and N. Then MN has length FIGURE

Accepted Answer

Firstly, the Pythagorean theorem gives align*AB &=AC^2+BC^2 &= 12^2+5^2 & =144+25 &=169 &= 13.align* Also, AM = AC = 12 and BN = BC = 5 since they are both radii of the respective circles. Thus MB = AB-AM = 13-12 = 1, and so MN = BN-BM = 5-1 = 4.