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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

35 mm
35 mm

23 mm
23 mm

14 mm

Answer 1

To find the length of \( FI \), we can apply the properties of tangents and triangles in a circle.

Given:

• \( AF = 28 \) mm (the radius).
• \( \angle FAJ = 39^\circ \).

Since \( FI \) is tangent to the circle at point \( F \), we know that it is perpendicular to the radius \( AF \) at point \( F \). Therefore, we can consider triangle \( AFI \):

1. In right triangle \( AFI \):

   • \( AF \) is the adjacent side to \( \angle FAJ \).
   • \( FI \) is the opposite side to \( \angle FAJ \).

2. We can use the tangent function: \[ \tan(\angle FAJ) = \frac{FI}{AF} \]

Substituting the known values: \[ \tan(39^\circ) = \frac{FI}{28} \]

Now, we calculate \( \tan(39^\circ) \):

• \( \tan(39^\circ) \approx 0.8098 \) (using a calculator).

Substituting this into the equation: \[ 0.8098 = \frac{FI}{28} \]

Now, solve for \( FI \): \[ FI = 28 \cdot 0.8098 \approx 22.7 \text{ mm} \]

Rounding off gives approximately \( 23 \) mm.

Thus, the measure of \( FI \) is:

23 mm.