Question

given that qm = 15 units, sm = 10 units, and rm = 18 units, what is the length of segment pm?
a. 12 units
b. 7 units
c. 13 units
d. 8 units

Explanation:

Step1: Recall the Intersecting Chords Theorem

The Intersecting Chords Theorem states that if two chords intersect in a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. Mathematically, if chords \(PQ\) and \(RS\) intersect at point \(M\), then \(QM \times PM = SM \times RM\).

Step2: Substitute the given values into the formula

We are given \(QM = 15\) units, \(SM = 10\) units, and \(RM = 18\) units. Let \(PM = x\). According to the theorem:

$$ QM \times PM = SM \times RM $$

Substituting the known values:

$$ 15 \times x = 10 \times 18 $$

Step3: Solve for \(x\)

First, calculate the right-hand side: \(10 \times 18 = 180\). Then, solve for \(x\) by dividing both sides of the equation by 15:

$$ x = \frac{180}{15} = 12 $$

Answer:

A. 12 units