Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.
More Regents problems.
Geometry Regents, June 2013
Part I: Each correct answer will receive 2 credits.
25. Which graph represents a circle whose equation is x2 + (y - 1)2 = 9?
Answer: 1) [SEE IMAGE]
The center of the circle is (0, 1) and the radius is 3. That is Choice (1).
The equation for a circle is (x - h)2 + (y - 1)2 = r2, where (h,k) is the center and r is the radius.
26. What is the perimeter of a rhombus whose diagonals are 16 and 30?
1) 92
2) 68
3) 60
4) 17
Answer: 2) 68
The diagonals of a rhombus will split the quadrilateral into 4 congruent right triangles. Since the diagonals are 16 and 30, then the legs of the right triangle will be 8 and 15.
Using Pythagorean Theorem, the hypotenuse, which is the side of the rhombus, will be 17. (You should know your Pythagorean Triples.)
The perimeter is 4 * 17 = 68.
27. In right triangle ABC shown in the diagram below, altitude BD is drawn to hypotenuse AC, CD = 12, and AD = 3.
What is the length of AB?
1) 5 SQRT(3)
2) 6
3) 3 SQRT(5)
4) 9
Answer: 3) 12
According to the Right Triangle Altitude Theorem, (AD)(DC) = (DB)2.
(DB)2 = (3)(12) = 36
DB = 6
But this isn't the side we are looking for.
Using Pythagorean Theorem, 32 + 62 = (AB)2
9 + 36 = (AB)2
45 = (AB)2
AB = SQRT(45) = SQRT(9 * 5) = SQRT(9)SQRT(5) = 3 SQRT(5)
More directly, you could have written the proportion:
AB / 15 = 3 / AB
... which would have led to the same answer.
28. Secants JKL and JMN are drawn to circle O from an external point, J.
If JK = 8, LK = 4, and JM = 6, what is the length of JN?
1) 16
2) 12
3) 10
4) 8
Answer: 1) 16
The Secant Secant Theorem tells us that the following products are equal:
If LK = 4 then JL = 8 + 4 = 12.
(8)(12) = (6)(JN)
96 = 6(JN)
JN = 16
End of Part I.
More to come. Comments and questions welcome.
More Regents problems.
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