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.