Answer:
A) 40 ft
B) 36.9°
Step-by-step explanation:
Part A
The tangent to a circle is perpendicular to the radius at the point of tangency.
As line segment KJ is tangent to circle C at point H, the radius CH is perpendicular to the tangent line KJ at point H. Therefore, angle CHK measures 90°, making triangle CHK a right triangle.
Given that CK = 50 ft and HK = 30 ft, we can determine the length of the radius CH by using the Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the legs of a right triangle:
[tex]CH^2+HK^2=CK^2\\\\CH^2+30^2=50^2\\\\CH^2+900=2500\\\\CH^2=1600\\\\CH=\sqrt{1600}\\\\CH=40[/tex]
Therefore, the length of the radius of the circle is 40 ft.
[tex]\dotfill[/tex]
Part B
To find the measure of angle KCH, we can use the sine trigonometric ratio. In this case, the side opposite the angle is HK and the hypotenuse is CK. Therefore:
[tex]\sin \theta=\dfrac{\textsf{Opposite side}}{\textsf{Hypotenuse}}\\\\\\\sin KCH=\dfrac{HK}{CK}\\\\\\\sin KCH=\dfrac{30}{50}\\\\\\\sin KCH=\dfrac{3}{5}\\\\\\KCH=\sin^{-1}\left(\dfrac{3}{5}\right)\\\\\\KCH=36.8698976458...^{\circ}\\\\\\KCH=36.9^{\circ}\; \sf (nearest\;tenth)[/tex]
Therefore, the measure of ∠KCH is 36.9° (rounded to the nearest tenth).
