Question

select the correct answer. what is the value of x in the given figure? figure not drawn to scale a. 8 b. 12 c. 13\frac{1}{3} d. 18

Explanation:

Step1: Recall the secant-secant theorem

The secant-secant theorem states that if two secant segments are drawn from a point outside a circle, then the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part. Let the external part of the first secant be \( 6 \), and the entire length be \( 6 + x \). The external part of the second secant is \( 8 \), and the entire length is \( 8 + 10 = 18 \).

Step2: Apply the theorem

Using the secant-secant theorem formula: \( 6(6 + x)=8\times18 \)
First, calculate the right - hand side: \( 8\times18 = 144 \)
Then, expand the left - hand side: \( 36+6x = 144 \)
Subtract \( 36 \) from both sides: \( 6x=144 - 36=108 \)
Divide both sides by \( 6 \): \( x=\frac{108}{6}=18 \)? Wait, no, wait. Wait, maybe I mixed up the segments. Wait, the two secants: one secant has external part \( 6 \) and internal part \( x \), so the whole length is \( 6 + x \). The other secant has external part \( 8 \) and internal part \( 10 \), so the whole length is \( 8+10 = 18 \). Wait, no, the correct formula is: If a secant from the external point has length \( a \) (external) and \( b \) (internal), and another secant has length \( c \) (external) and \( d \) (internal), then \( a(a + b)=c(c + d) \). Wait, in the figure, the two secants: one secant is composed of the external segment \( 6 \) and the internal chord segment \( x \), so the total length of the secant is \( 6 + x \). The other secant is composed of the external segment \( 8 \) and the internal chord segment \( 10 \), so the total length of the secant is \( 8+10 = 18 \). Wait, no, maybe the external part of the first secant is \( 6 \), and the internal part is \( x \), and the external part of the second secant is \( 8 \), and the internal part is \( 10 \). Then according to the secant - secant rule: \( 6\times(6 + x)=8\times(8 + 10) \)
Wait, \( 8\times(8 + 10)=8\times18 = 144 \)
\( 6\times(6 + x)=144 \)
\( 6 + x=\frac{144}{6}=24 \)
\( x = 24-6 = 18 \)? But that's option D. But wait, maybe I made a mistake. Wait, maybe the two secants: one secant has external part \( 8 \) and internal part \( 10 \), and the other has external part \( 6 \) and internal part \( x \). Wait, no, let's re - examine the figure. The two secant segments: from the external point, one secant goes through the circle, with the part outside the circle being \( 6 \) and the part inside the circle (the chord) being \( x \). The other secant goes through the circle, with the part outside the circle being \( 8 \) and the part inside the circle (the chord) being \( 10 \). The secant - secant theorem is: (length of external part of first secant)×(length of external part + length of internal part of first secant)=(length of external part of second secant)×(length of external part + length of internal part of second secant)
So, \( 6\times(6 + x)=8\times(8 + 10) \)
\( 6\times(6 + x)=8\times18 \)
\( 6\times(6 + x)=144 \)
\( 6 + x = 24 \)
\( x=18 \)
Wait, but let's check again. Wait, maybe the formula is (external part)×(whole secant)=(external part)×(whole secant). So for the first secant: external = 6, whole = 6 + x. For the second secant: external = 8, whole = 8+10 = 18. So \( 6(6 + x)=8\times18 \)
\( 36+6x = 144 \)
\( 6x=108 \)
\( x = 18 \)

Answer:

D. 18