enigma123 wrote:
In the figure, circle O has center O, diameter AB and a radius of 5. Line CD is parallel to the diameter. What is the perimeter of the shaded region?
A. \((\frac{5}{3})\pi + 5\sqrt{3}\)
B. \((\frac{5}{3})\pi + 10\sqrt{√}3\)
C. \((\frac{10}{3})\pi + 5\sqrt{3}\)
D. \((\frac{10}{3})\pi + 10\sqrt{3}\)
E. \((\frac{10}{3})\pi + 20\sqrt{3}\)
IMPORTANT: unless stated otherwise, the
diagrams in Problem Solving geometry questions are DRAWN TO SCALE.
I'll solve this question using
estimation.
Since the diameter AB = 10, we can ESTIMATE the length of CB.
It looks like CB is just a little bit shorter than AB.
So, I'll say that the length of side CB is approximately 9.
This means the length of side EB is approximately 9 as well.
Finally, arc EC looks a little bit shorter than sides CB and EB, so I'll estimate it to be length 8
So, the TOTAL perimeter = 9 + 9 + 8 = 26
Now check the answer choices:
ASIDE: On test day, everyone should know the following apprximations:
√2 ≈ 1.4
√3 ≈ 1.7
√5 ≈ 2.2
Also, we'll say that pi ≈ 3A. (5/3)pi + 5√3 ≈ 5 + 8.5 ≈ 13.5
B. (5/3)pi + 10√3 ≈ 5 + 17 ≈ 22
C. (10/3)pi + 5√3 ≈ 10 + 8.5 ≈ 18.5
D. (10/3)pi + 10√3 ≈ 10 + 17 ≈ 27
E. (10/3)pi + 20√3 ≈ 10 + 34 ≈ 44
Of these, it appears that D is the closest.
Aside: We can see that answer choice B is pretty close too. At this point, you have a time-management decision. You can either stick with D, and use your extra time elsewhere, or your can spend time trying to be more certain of the answer. Your choice.
That said, D is the correct answer.
Cheers,
Brent
_________________