Find the missing side lengths
1 answer:
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<h3>Answer: A) 6.4</h3>
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Explanation:
Refer to the diagram below. I've added the points A, B, C, D, and E. Point A is the center of the circle while points B, C and D are along the circle's edge. The radius AC and chord BD intersect at point E.
Recall that whenever we have a radius perpendicular to a chord like this, it means that the radius bisects the chord. So that means chord BD is cut into two equal pieces BE and ED. The diagram given to you tells us that ED = x, so BE is also x units long as well. Overall, BD = BE+ED = x+x = 2x units.
Since BE = ED, the arcs formed are congruent as well. Minor arc BC is the same measure as minor arc CD. Both are 45 degrees. They combine to 90 degrees. In turn, this leads to central angle DAB to be 90 degrees.
Note how triangle DAB is an isosceles right triangle. The two equal legs are the two radii AB and AD, both which are 9 units long. The hypotenuse BD is equal to sqrt(2) times the leg length, so BD = 9*sqrt(2) units long.
Earlier I mentioned that BD = 2x also. We'll equate the two right hand side expressions and solve for x like so:
2x = 9*sqrt(2)
x = (9*sqrt(2))/2
x = (9/2)*sqrt(2)
x = 4.5*sqrt(2)
x = 6.36396103067892
That value is approximate. Rounding to the nearest tenth leads to the final answer of 6.4; therefore, the answer is choice A.