Question

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kylie writes an equation to solve for the diameter, d, of the circle shown. solve kylie’s equation to find the diameter of the circle. 4 (d + 4) = 7² what is the diameter? options: 7.5 in., 8.25 in., 11.25 in., 13.25 in.

Explanation:

Step1: Divide both sides by 4

To isolate the term with \( d \), we divide both sides of the equation \( 4(d + 4)=7^{2} \) by 4. First, calculate \( 7^{2}=49 \), so the equation becomes \( 4(d + 4)=49 \). Then dividing both sides by 4: \( \frac{4(d + 4)}{4}=\frac{49}{4} \), which simplifies to \( d + 4 = 12.25 \)? Wait, no, wait the original equation—wait, maybe I misread. Wait the user's equation: looking at the image, the equation is \( 4(d + 4)=7^{2} \)? Wait no, the image says "4 (d + 4) = 7²"? Wait no, 7 in the diagram, maybe it's \( 4(d + 4)=7^{2} \)? Wait 7 squared is 49. Wait but let's check again. Wait the problem: Kylie's equation is \( 4(d + 4)=7^{2} \)? Wait no, maybe it's \( 4(d + 4)=7^2 \)? Wait 7 squared is 49. Then:

Wait, no, maybe the equation is \( 4(d + 4)=7^2 \)? Wait 7*7=49. Then:

Step1: Divide both sides by 4

Given \( 4(d + 4)=49 \) (since \( 7^2 = 49 \)), divide both sides by 4:
\( d + 4=\frac{49}{4} \)
\( d + 4 = 12.25 \)? Wait that can't be, because the options are 7.5, 8.25, 11.25, 13.25. Wait maybe the equation is \( 4(d + 4)=7^2 \) is wrong. Wait maybe the equation is \( 4(d + 4)=7^2 \)? No, wait the diagram has 4 in and 7 in. Wait, maybe the equation is \( 4(d + 4)=7^2 \)? Wait no, let's re-express. Wait the correct equation from the power of a point: if a tangent and a secant are drawn from a point outside the circle, then \( \text{tangent}^2=\text{external part} \times \text{whole secant} \). So the tangent is 7, the external part is 4, and the whole secant is \( 4 + d \) (since \( d \) is the diameter, so the radius is \( d/2 \), but wait, the secant goes from the external point through the circle: the external segment is 4, and the internal segment is the diameter \( d \)? Wait no, the secant length is external part + internal part. Wait, the power of a point formula: \( t^2 = e \times (e + i) \), where \( t \) is tangent, \( e \) is external segment, \( i \) is internal segment. So here, tangent is 7, external segment is 4, internal segment is \( d \) (diameter). So \( 7^2 = 4 \times (4 + d) \), which is \( 49 = 4(d + 4) \), which is the equation. Then solving:

Step1: Divide both sides by 4

\( \frac{49}{4}=d + 4 \)
\( 12.25 = d + 4 \)

Step2: Subtract 4 from both sides

Subtract 4 from both sides to solve for \( d \):
\( d = 12.25 - 4 \)
\( d = 8.25 \)? Wait no, 12.25 - 4 is 8.25? Wait 12.25 - 4 is 8.25? Wait 12.25 - 4 = 8.25? Wait 12 - 4 = 8, 0.25 - 0 = 0.25, so 8.25. Wait but the options have 8.25 as an option. Wait let's check the equation again. Wait maybe the equation is \( 4(d + 4)=7^2 \), so 4(d + 4)=49, then d + 4=49/4=12.25, then d=12.25 -4=8.25. Yes, that's correct.

Wait, but let's do the steps properly:

Step1: Expand or divide? Let's divide both sides by 4.

Given equation: \( 4(d + 4) = 7^2 \)
Calculate \( 7^2 = 49 \), so:
\( 4(d + 4) = 49 \)

Divide both sides by 4:
\( d + 4 = \frac{49}{4} \)
\( d + 4 = 12.25 \)

Step2: Subtract 4 from both sides.

\( d = 12.25 - 4 \)
\( d = 8.25 \)

Answer:

8.25 in.