Saturday, October 30, 2021

Geometry Problems of the Day (Geometry Regents, January 2012)

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, January 2012

1. Line n intersects lines l and m, forming the angles shown in the diagram below.

Which value of x would prove l || m?

1) 2.5
2) 4.5
3) 6.25
4) 8.75

If the two lines are parallel, then the Altenate Interior Angles must be congruent. So write an equation that says that they equal each other:

18x - 12 = 6x + 42
12x = 54
x = 54/12 = 4.5

You also could have plugged in each choice into the two expressions to find which value made them equal.

2. In a given triangle, the point of intersection of the three medians is the same as the point of intersection of the three altitudes. Which classification of the triangle is correct?

1) scalene triangle
2) isosceles triangle
3) equilateral triangle
4) right isosceles triangle

If choice (4) were correct, then Choice (2) would also be correct. So we can eliminate Choice (4). Also, the altitudes in a right triangle meet at the right angle. Medians don't meet at any of the vertices.

If the centroid and the orthocenter are the same point, then the triangle must be an equilateral triangle where each altitude is also a perpendicular bisector and an angle bisector.

3. A circle has the equation (x - 2)2 + (y + 3)2 = 36. What are the coordinates of its center and the length of its radius?

1) (-2,3) and 6
2) (2,-3) and 6
3) (-2,3) and 36
4) (2,-3) and 36

The equation of a circle is given by the formula (x - h)2 + (y - k)2 = r2, where (h,k) in the center of the circle and r is the radius. Note that there are MINUS signs in the formula, so the signs will be flipped.

If r2 = 36, then the radius is 6, not 36. Eliminate Choices (3) and (4).

The signs are flipped, so the center if (2,-3), which is Choice (2).

4. In the diagram below, MATH is a rhombus with diagonals AH and MT.
If m∠HAM = 12, what is m∠AMT?
1) 12
2) 78
3) 84
4) 156

The diagonals of a rhombus for four right triangles. So in each triangle, the two acute angles are complementary, adding up to 90 degress.

So 90 - 12 = 78.

5. A line segment has endpoints (4,7) and (1,11). What is the length of the segment?

1) 5
2) 7
3) 16
4) 25

Use the distance formula or Pythagorean Theorem.

d = √( (4 - 1)2 + (11 - 7)2 ) = √( (3)2 + (4)2 )
= √(9 + 16) = √(25) = 5

If you know your Pythagorean Triples, you should have gotten that long before the end.

More to come. Comments and questions welcome.

More Regents problems.

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