2018 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 16 Medium

Right triangle ABC has leg lengths AB=20 and BC=21. Including \overline{AB} and \overline{BC}, how many line segments with integer length can be drawn from vertex B to a point on hypotenuse \overline{AC}? (2018 AMC 10A Problem, Question#16)

  • A.

    5

  • B.

    8

  • C.

    12

  • D.

    13

  • E.

    15

Answer:D

As the problem has no diagram, we draw a diagram. The hypotenuse has length 29. Let P be the foot of the altitude from B to AC. Note that BP is the shortest possible length of any segment. Writing the area of the triangle in two ways, we can solve for BP=\frac{20\cdot 21}{29}, which is between 14 and 15.

Let the line segment be BX, with X on AC. As you move X along the hypotenuse from A to P, the length of BX strictly decreases, hitting all the integer values from 20,19,\cdots ,15(IVT). Similarly, moving X from P to C hits all the integer values from 15,16,\cdots ,21. This is a total of \boxed{\rm (D)~13} line segments.

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