A park maintenance person stands 23 m from a circular monument. If you assume her lines of site are tangents to the monument and make an angle of 23degrees, what is the measure of the monument that her lines of site intercect?
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To find the measure of the monument that the lines of sight intersect, we need to determine the angle at the center of the circle.
Step 1: Draw a diagram of the situation described. The monument is represented as a circle, and the park maintenance person's viewpoint is represented by a point outside the circle, with two lines of sight (tangents) drawn from that point to the circle.
Step 2: From the given information, we know that the distance between the park maintenance person and the circle is 23 meters. This forms a right triangle with the radius of the circle and the line connecting the center of the circle to the point of tangency.
Step 3: To find the length of the radius, we can use the tangent of the angle formed between the radius and the line connecting the center of the circle to the point of tangency. Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle.
In this case, the opposite side is the distance between the park maintenance person and the circle (23 meters), and the adjacent side is the radius (which we need to find). So, the tangent of the angle is given by the formula:
tangent = opposite/adjacent
tangent(23 degrees) = 23/adjacent
Step 4: Solve for the adjacent side (radius). Rearrange the formula to solve for the adjacent side:
adjacent = opposite / tangent(23 degrees)
adjacent = 23 / tangent(23 degrees)
Using a scientific calculator, find the tangent of 23 degrees and perform the division to find the value of the adjacent side (radius).
Step 5: Once you have the value for the radius, double it to get the diameter of the circle. The diameter represents the measure of the monument that the lines of sight intersect.