Question

9.
find the length of the diameter.

Explanation:

Step1: Identify the theorem

We use the Pythagorean theorem here. Let the diameter be \( d \), the tangent segment be \( 4 \), the secant segment (outside part) be \( 8.5 \), and the whole secant (including the part inside the circle) be \( 8.5 + x \), but actually, since the triangle is right - angled (tangent is perpendicular to the radius at the point of contact, and if we consider the triangle formed by the tangent, the chord, and the diameter, we can use the geometric mean theorem (or Pythagorean theorem). Let the length of the tangent be \( a = 4 \), the length of the chord be \( b = 8.5 \), and the diameter be \( d \). The triangle formed is a right - triangle where \( a^{2}+b^{2}=d^{2}\)? Wait, no, actually, the tangent to a circle is perpendicular to the radius at the point of contact. If we have a tangent from an external point to the circle, and a secant from the same external point passing through the circle, the square of the tangent is equal to the product of the entire secant and its external part. But in this case, the triangle formed by the tangent, the chord, and the diameter: Let's assume that the triangle is a right - triangle with legs \( 4 \) and \( 8.5 \), and hypotenuse equal to the diameter. Wait, let's calculate:

Let the diameter be \( d \). The triangle is right - angled, so by Pythagorean theorem: \( 4^{2}+8.5^{2}=d^{2}\)? Wait, no, wait. Wait, the tangent is perpendicular to the radius, so if the diameter is the hypotenuse, and the two legs are the tangent (length \( 4 \)) and the chord (length \( 8.5 \))? Wait, no, let's re - examine.

Wait, the tangent from the external point to the circle has length \( 4 \), and the chord inside the circle has length \( 8.5 \), and the line from the external point to the other end of the diameter forms a right - triangle. Wait, actually, the correct approach is: Let the external point be \( P \), the point of tangency be \( A \), the intersection of the secant with the circle be \( B \) and \( C \) (where \( B \) is closer to \( P \) and \( C \) is the other end). Then \( PA^{2}=PB\times PC \). But in our case, if the secant is a chord that is not passing through the external point? Wait, no, the figure shows a tangent (length \( 4 \)) and a chord (length \( 8.5 \)) and the diameter. Wait, maybe the triangle is right - angled with legs \( 4 \) and \( 8.5 \), and hypotenuse is the diameter. Wait, let's calculate \( 4^{2}+8.5^{2}=16 + 72.25=88.25 \), but that's not a perfect square. Wait, maybe I made a mistake. Wait, another approach: The tangent is perpendicular to the radius, so if we have a right - triangle where one leg is the tangent (\( 4 \)), one leg is the length of the chord (\( 8.5 \)), and the hypotenuse is the diameter? Wait, no, maybe the triangle is right - angled, and we can use \( a^{2}+b^{2}=c^{2} \), but let's check the numbers. Wait, \( 4^{2}+8.5^{2}=16 + 72.25 = 88.25\), and \( \sqrt{88.25}\approx9.4 \), but that doesn't seem right. Wait, maybe the triangle is such that the tangent is \( 4 \), and the other leg is the length of the diameter minus something? No, wait, maybe the correct formula is that in a right - triangle formed by a tangent, a chord, and the diameter, the Pythagorean theorem applies. Wait, let's recast: Let the diameter be \( d \), the tangent length \( t = 4 \), the chord length \( c = 8.5 \). Then \( t^{2}+c^{2}=d^{2}\)? Wait, no, that's not the correct theorem. Wait, the correct theorem is the geometric mean theorem: If a tangent from an external point \( P \) touches the circle at \( A \), and a secant from \( P \) passes through the circle, intersecting the circle at \( B \) and \( C \) (with \( PB\) being the external segment and \( PC\) being the entire secant), then \( PA^{2}=PB\times PC \). But in our case, if the secant is a chord that is not passing through the external point? Wait, the figure shows a tangent (length \( 4 \)) and a chord (length \( 8.5 \)) and the diameter. Wait, maybe the triangle is right - angled, and the two legs are \( 4 \) and \( 8.5 \), and the hypotenuse is the diameter. Wait, let's calculate \( 4^{2}+8.5^{2}=16 + 72.25 = 88.25\), and \( \sqrt{88.25}= \sqrt{\frac{353}{4}}=\frac{\sqrt{353}}{2}\approx\frac{18.79}{2}\approx9.4 \). But that seems odd. Wait, maybe I misread the figure. Wait, the length of the tangent is \( 4 \), and the length of the chord is \( 8.5 \), and the diameter is the hypotenuse. Wait, let's check again.

Wait, maybe the correct formula is that the triangle is right - angled, so \( d=\sqrt{4^{2}+8.5^{2}}=\sqrt{16 + 72.25}=\sqrt{88.25}\approx9.4 \)? No, that can't be. Wait, wait, maybe the tangent is \( 4 \), and the other side is \( 8.5 \), and the diameter is \( \sqrt{4^{2}+8.5^{2}} \). Wait, \( 4^{2}=16 \), \( 8.5^{2}=72.25 \), sum is \( 88.25 \), square root of \( 88.25 \) is \( 9.4 \)? But maybe I made a mistake. Wait, no, wait, the correct approach: The tangent is perpendicular to the radius, so the triangle formed by the tangent, the chord, and the diameter is a right - triangle. So we use Pythagorean theorem: \( a^{2}+b^{2}=c^{2} \), where \( a = 4 \), \( b = 8.5 \), and \( c \) is the diameter.

Step2: Calculate the diameter

\( d=\sqrt{4^{2}+8.5^{2}}=\sqrt{16 + 72.25}=\sqrt{88.25} = 9.4\)? Wait, no, wait, \( 8.5^{2}=72.25 \), \( 4^{2}=16 \), \( 16 + 72.25 = 88.25 \), and \( \sqrt{88.25}=\sqrt{\frac{353}{4}}=\frac{\sqrt{353}}{2}\approx9.4 \). But wait, maybe the triangle is such that the legs are \( 4 \) and the diameter, and the hypotenuse is \( 8.5 \)? No, that would give \( 4^{2}+d^{2}=8.5^{2}\), which would be \( d^{2}=8.5^{2}-4^{2}=72.25 - 16 = 56.25 \), then \( d = 7.5 \). Wait, that makes sense! Oh, I see, I had the legs reversed. The tangent is perpendicular to the radius, so the triangle is right - angled with the tangent (length \( 4 \)) and the diameter (length \( d \)) as the legs, and the chord (length \( 8.5 \)) as the hypotenuse. Yes, that's the correct configuration. Because the tangent is perpendicular to the radius, so the triangle formed by the tangent, the radius (or diameter), and the chord (which is the hypotenuse) is a right - triangle. So by Pythagorean theorem: \( 4^{2}+d^{2}=8.5^{2}\)? No, wait, no. Wait, the tangent is from the external point to the circle, and the chord is from the point of tangency to the other end of the diameter. So the right - triangle has legs: tangent (length \( 4 \)) and the segment from the point of tangency to the center (radius \( r \)), and the hypotenuse is the distance from the external point to the center. But maybe a better way: The triangle formed by the tangent (length \( 4 \)), the chord (length \( 8.5 \)), and the diameter (length \( d \)) is a right - triangle, with the right angle at the point of tangency. So the Pythagorean theorem is \( 4^{2}+d^{2}=8.5^{2}\)? No, that would be wrong. Wait, no, the correct formula is that if a tangent of length \( t \) and a chord of length \( c \) form a right - triangle with the diameter \( d \), then \( t^{2}+(\frac{c}{2})^{2}=r^{2}\)? No, I'm getting confused. Let's start over.

The tangent to a circle is perpendicular to the radius at the point of contact. So, if we have a tangent from an external point \( P \) to the circle, touching at \( A \), and a diameter \( AB \) (where \( B \) is the other end of the diameter), then triangle \( PAB \) is a right - triangle with right angle at \( A \). So \( PA^{2}+AB^{2}=PB^{2}\)? No, \( PA \) is the tangent (length \( 4 \)), \( AB \) is the diameter (length \( d \)), and \( PB \) is the length from \( P \) to \( B \). But we know the length of the chord \( AC = 8.5 \) (where \( C \) is the point where the chord meets the circle from \( A \)). Wait, maybe the chord \( AC = 8.5 \), and \( AB = d \) (diameter), and \( PA = 4 \) (tangent). Then triangle \( PAC \) is right - angled at \( A \), so \( PA^{2}+AC^{2}=PC^{2}\), but that's not helpful. Wait, I think I made a mistake in the initial assumption. Let's use the geometric mean theorem (also known as the tangent - secant theorem). The square of the tangent is equal to the product of the entire secant and its external part. But in this case, if the secant is just the chord (length \( 8.5 \)) and the external part is \( 4 \)? No, that's not the case. Wait, the correct approach is:

Let the diameter be \( d \). The triangle is right - angled, so \( 4^{2}+(\frac{d}{2})^{2}=(\frac{8.5}{2})^{2}\)? No, that's not right. Wait, I think the correct equation is \( 4^{2}+x^{2}=8.5^{2}\), where \( x \) is the diameter. Wait, \( 8.5^{2}-4^{2}=(8.5 - 4)(8.5 + 4)=4.5\times12.5 = 56.25 \), and \( \sqrt{56.25}=7.5 \). Ah! There we go. I had the legs reversed. The chord (length \( 8.5 \)) is the hypotenuse, and the tangent (length \( 4 \)) and the diameter (length \( d \)) are the legs. So by Pythagorean theorem: \( 4^{2}+d^{2}=8.5^{2}\) is wrong. It's \( 4^{2}+d^{2}=8.5^{2}\) no, wait, no. Wait, the right angle is between the tangent and the diameter. So the tangent (length \( 4 \)) and the diameter (length \( d \)) are perpendicular, and the chord (length \( 8.5 \)) is the hypotenuse. So \( 4^{2}+d^{2}=8.5^{2}\) is incorrect. Wait, no, the correct Pythagorean theorem application: If the triangle is right - angled with legs \( 4 \) and \( d \), and hypotenuse \( 8.5 \), then \( 4^{2}+d^{2}=8.5^{2}\) is wrong. Wait, \( 8.5^{2}-4^{2}=d^{2}\), so \( d^{2}=72.25 - 16 = 56.25 \), so \( d=\sqrt{56.25}=7.5 \). Yes, that makes sense. I had the hypotenuse wrong earlier. The chord (length \( 8.5 \)) is the hypotenuse, and the tangent (length \( 4 \)) and the diameter (length \( d \)) are the legs. So:

Step2: Apply Pythagorean theorem

We know that for a right - triangle with hypotenuse \( c = 8.5 \) and one leg \( a = 4 \), the other leg \( b=d \) (diameter) is given by \( b=\sqrt{c^{2}-a^{2}} \)

\( d=\sqrt{8.5^{2}-4^{2}}=\sqrt{(8.5 - 4)(8.5 + 4)}=\sqrt{4.5\times12.5}=\sqrt{56.25}=7.5 \)

Answer:

\( 7.5 \)