Thursday, October 14, 2021

Geometry Problems of the Day (Geometry Regents, August 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, August 2012

Part I: Each correct answer will receive 2 credits.

26. Which type of triangle can be drawn using the points (-2,3), (-2,-7), and (4,-5)?

1) scalene
2) isosceles
3) equilateral
4) no triangle can be drawn

Answer: 2) isosceles

The quickest way to answer the question would be to sketch it on the graph paper provided and just look at it.

The only way for NO triangle to be possible would be if all three points lined up on a straight line. But (4, -5) is not on the same line as the other two points, so there is a triangle.

For this to be an isosceles triangle, we need to know the lengths of the three sides. The first side is a vertical line with a length of 10. The other two sides need the Distance Formula or Pythagorean Theorem.

SQRT( (-2 - 4)2 + (3 - -5)2) = SQRT(62 + 82) = SQRT(100) = 10. The triangle is AT LEAST isosceles. Check if it is equilateral

SQRT( (-2 - 4)2 + (-7 - -5)2) = SQRT(62 + (-2)2) = SQRT (68). It is NOT equilateral.

One could argue that since the question didn't ask "which best describes" the triangle that the answer could not be equilateral because then it would also be isosceles.

Also, it is easy to rule out equilateral because it is NOT possible to have an equilateral triangle where all the coordinates of all the vertices are whole numbers.

27. In the diagram below, DE joins the midpoints of two sides of triangle ABC.
Which statement is not true?

1) CE = 1/2 CB
2) DE = 1/2 AB
3) area of triangle CDE = 1/2 area of triangle CAB
4) perimeter of triangle CDE = 1/2 perimeter of triangle CAB

Answer: 3) area of triangle CDE = 1/2 area of triangle CAB

The area of triangle CDE will be one quarter of the area of triangle CAB. The base is half of the larger triangle and the height is half of the larger triangle. One half of one half is one quarter, not one half.

All the other choices involve LENGTH, not AREA. All of the lengths are half, which means that the perimeter, which is the sum of the length, is half.

28. Which equation represents the line that is perpendicular to 2y = x + 2 and passes through the point (4,3)?

1) y = 1/2 x - 5
2) y = 1/2 x + 1
3) y = -2x + 11
4) y = -2x - 5

Answer: 3) y = -2x + 11

Divide the original equation by 2 and you get y = 1/2 x + 1, so the slope of the original equation is 1/2. Any line perpendicular to it will have a slope of -2, not 1/2. Eliminate Choices (1) and (2).

Substitute 4 for x:
y = -2(4) + 11 = -8 + 11 = 3 --> (4,3) is a solution to this line. This is the correct choice.
y = -2(4) - 5 = - 8 - 5 = -13 --> (4,3) is NOT a solution to this line. Eliminate this choice.

End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

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