Secant line MA←→− of circle G intersects secant line TH←→ outside the circle at point P . Point M is between point A and point P . Point T is between point H and point P . Determine the measure of PT¯¯¯¯¯¯¯ if PM=6 , MA=10 , and PH=24 .(1 point) PT

Answer 1

To solve for the length of \( PT \), we can use the properties of secant segments in a circle. According to the secant-secant power theorem, the product of the segments of one secant line is equal to the product of the segments of the other secant line.

We have the following points and segments:

• Let \( PM = 6 \)
• Let \( MA = 10 \) (thus, \( PA = PM + MA = 6 + 10 = 16 \))
• Let \( PH = 24 \)

Now, we denote \( PT \) as \( x \). The entirety of petal \( PH \) consists of \( PT + TH \), and since we don't have \( TH \), we focus just on the segments that are known for the two secants intersecting at point \( P \).

Using the Secant-Secant Power Theorem: \[ (PM)(PA) = (PT)(PH) \] Substituting the known values: \[ (6)(16) = (x)(24) \] Simplifying the left side: \[ 96 = 24x \] To find \( x \), we divide both sides by 24: \[ x = \frac{96}{24} = 4 \] Thus, the length of \( PT \) is: \[ \boxed{4} \]