Use the image to answer the question.

Question

A line with arrow heads at both its ends tangentially intersects the circumference of a circle at a single marked point labeled upper F. The center of the circle is marked by point upper A, which connects to a separate circumference point: upper J.



If FI
is tangent to the circle A
at point F
, AF=28
mm, and ∠FAJ=39
degrees, then what is the measure of FI
?

(1 point)
Responses

21 mm
21 mm

23 mm
23 mm

14 mm
14 mm

35 mm

Answer 1

To find the length of line segment FI, which is tangent to the circle at point F, we can use the properties of tangents and right triangles.

Given that:

• AF = 28 mm (the radius of the circle),
• ∠FAJ = 39 degrees.

Using these details, we can find the measure of the tangent FI using trigonometry.

In triangle AIF:

• FI is the tangent (which is perpendicular to the radius AF at point F),
• Thus, triangle AIF is a right triangle,
• Therefore, we can use the tangent function.

Using the sine function in triangle AIF: \[ \sin(\angle FAJ) = \frac{FI}{AF} \] Which leads to: \[ FI = AF \cdot \sin(\angle FAJ) \] Substituting the known values: \[ FI = 28 \cdot \sin(39^\circ) \]

Using a scientific calculator, we find: \[ \sin(39^\circ) \approx 0.6293 \] Now substituting: \[ FI = 28 \cdot 0.6293 \approx 17.6 \text{ mm} \]

However, the responses available are specific values: 21 mm, 23 mm, 14 mm, and 35 mm. The calculated value suggests that either there is further rounding or approximation in how the answer choices were developed.

Given that 17.6 mm is closest to 21 mm and no other option fits closely, we'll choose:

21 mm as the measure of FI.