Line segment QP is tangent to the circle. A circle is shown. Secant M P and tangent Q P intersect at point P outside of the circle. Secant M P intersects the circle at point N. The length of Q P is n, the length of N P is 11.5, and the length of M N is 24. What is the length of line segment QP? Round to the nearest unit. 13 units 17 units 18 units 20 units

The length of line segment QP is 20 units ⇒ 4th answer

Step-by-step explanation:

If a secant and a tangent are drawn from a point outside the circle, then the product of the lengths of the secant and its external segment equals the square of the length of the tangent segment

Look to the attached figure

∵ PQ is a tangent to the circle

∵ PM is a secant intersects the circle at points N and M

- That means the product of the lengths of PM and PN is

   equal to the square of the length of PQ

∴ (PQ)² = (PN). (PM)

∵ The length of Q P is n units

∴ PQ = n

∵ The length of N P is 11.5 units

∴ NP 11.5

∵ The length of M N is 24 units

∴ MN = 24

- The length of the secant PM is the sum of the lengths of PN

   and MN

∵ PM = PN+ NM

∴ PM = 11.5 + 24 = 35.5

Substitute the values of PQ, PN, and PM in the formula above

∵ n² = 11.5 × 35.5

∴ n² = 408.25

- Take √  for both sides

∴ n = 20.205197

- Round it to the nearest unit

∴ n = 20

∵ n is the length of PQ

∴ The length of line segment QP is 20 units

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