Thursday, June 01, 2017

Geometry Problems of the Day

The following problems were taken from the GEOMETRY (COMMON CORE) Regents Exam given on Thursday, January 26, 2017.
Previous problems can be found here.

Part 1

21. In the diagram below of circle O, GO 8 and m∠GOJ = 60°. What is the area, in terms of π, of the shaded region?

(4) 160π / 3.
Since 60° is 1/6th of the 360° degree in the complete circle, then the unshaded region of the circle is (1/6) πr2 = (1/6) π82
and the shaded portion would be (5/6) πr2 = (5/6) π82 = (5 * 64π) / 6 = (5 * 32π) / 3 = 160π/ 3.



22. A circle whose center is the origin passes through the point (5,12).
Which point also lies on this circle?

(3) (11, 2 sqrt(12))
The equation of the circle is x2 + y2 = r2. We can find r using the Distance Formula or Pythagorean Theorem: 52 + 122 = r2.
25 + 144 = 169 = r2
r = 13 (which you really should have known. Look up Pythagorean triples.)

Which of the other points creates a right triangle with a hypotenuse of 13?
(10, 3) definitely do not -- they don't even create a triangle with a side of 13. (-12, 13) can't because the hypotenuse is longer than the legs (plus it would have to be -12 and either 5 or -5).
Check 112 + (2sqrt(12))2 = 121 + 4(12) = 169 = 132. Looks good.
Check (-8)2 + (5sqrt(21))2 = 64 + 25(21) = way too much. No good.




Continue to the next problems.

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