Momtaj K. answered • 08/05/19
A Student Friendly Math Teacher
Question: In the diagram, triangle EFJ is inscribed in Circle C such that segment EF is a diameter of the circle, and the angle measure of JEF=30 degrees.
If the diameter, segment EF, has a length of 4 inches, what are the lengths of segments EJ and JF?
Select two answer choices, one for the length of segment EJ, and one for the length of segment JF.
Answer: We will be using trigonometry and a circle theorem to solve for the lengths of EJ and JF.
Step 1: Identifying the right angle. Arc EF measures 180 degrees because it is the arc of a semicircle. Arc EF is the intercepted arc of angle EJF. Inscribed angles are half the measure of their intercepted arc. So angle EJF is half the measure of its intercepted arc (arc EF). Since arc EF is 180 degrees, angle EJF is 180/2= 90 degrees.This proves that triangle EFJ is a right triangle which let's us use trigonometry to find the lengths of the sides.
Step 2: Find the length of EJ using the information that angle JEF is 30 degrees and that EF is 4 inches.
EF is the hypotenuse because it is opposite to the right angle. EJ is adjacent to the angle JEF. The trigonometric ratio that uses adjacent and hypotenuse is cos.
cos θ = adjacent / hypotenuse
So, cos (30) = EJ/4
4* √3/2 = EJ/4 *4
2√3 = EJ
Step 3: Find the length of JF using the information that angle JEF is 30 degrees and that EF is 4 inches.
EF is the hypotenuse because it is opposite to the right angle. JF is opposite to the angle JEF. The trigonometric ratio that uses opposite and hypotenuse is sin.
sin θ = opposite/ hypotenuse
So, sin (30) = JF/ 4
4* 1/2 = JF/4 *4
2 = JF
So JF = 2 inches and EJ = 2√3 inches.
