Geometry Word Problem

Question

a circular stone has a diameter of 250 feet. a passage 62 feet long leads toward the center of the stone. find the perpendicular distance x from the end of the passage to either side of the stone

Answer 1

Draw a diagram!

 

The circular stone has a diameter of 250 feet (and therefore a radius of 125 feet).

 

The passageway starts on the edge of the circle, leads toward the center of the stone, and is 62 feet long.  That leaves 63 (that is, 125-62) for the rest of the way to the center.

 

Now, the distance x from the end of the passageway to either side of the stone forms a right triangle with:

      side 63

      side x

      hypotenuse 125      (the radius of the circle)

 

From the Pythagorean Theorem:

     63² + x² = 125²

 

       x² = 125² - 63²

       x² = 15625 - 3969

       x = √11656

       x = 107.963

   x is approximately 108 feet

Answer 2

Treating this as the intersection of two chords, one chord is divided into two equal segments of length p, from center of passageway (at its end 62 ft from the circle). The chord of length 2p divides the diameter into two parts: one is 62 ft, the other is 250 - 62, or 188 ft.

 

Using the rule (theorem) that applies to intersecting chords, we have p•p = 62•188

or p² = 5456.  Therefore p = √5456 ≈ 73.865 ft.