Tuesday, November 09, 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

25. The coordinates of the endpoints of AB are A(0,0) and B(0,6). The equation of the perpendicular bisector of AB is

1) x = 0
2) x = 3
3) y = 0
4) y = 3

AB is a vertical line. The perpendicular bisector will therefore be a horizontal line. Note that this is the one exception to the inverse reciprocal rule. Eliminate choices (1) and (2).

The midpoint between (0,0) and (0,6) is (0,3). The horizontal line that crosses through that point is y = 3.

26. In the diagram below, point P is the centroid of triangle ABC.

If PM = 2x + 5 and BP = 7x + 4, what is the length of PM?

1) 9
2) 2
3) 18
4) 27

The medians of a triangle meet at a centroid. The centroid will divide the median into two segment with a ratio of 2:1. So BP = 2 * PM.

Therefore,

7x + 4 = 2(2x + 5)
7x + 4 = 4x + 10
3x = 6
x = 2

If x = 2, the PM = 2x + 5 = 2(2) + 5 = 9.

27. In triangle PQR, ∠PRQ is a right angle and RT is drawn perpendicular to hypotenuse PQ. If PT = x, RT = 6, and TQ = 4x, what is the length of PQ?

1) 9
2) 12
3) 3
4) 15

THe Right Triangle Altitude Theorem tells us that (PT)(TQ) = (RQ)2 because similar triangles are proportional.

(x)(4x) = 62
4x2 = 36
x2 = 9
x = 3

PQ = PT + TQ = x + 4x = 5x = 5(3) = 15

28. In triangle ABC, AB = 5 feet and BC = 3 feet. Which inequality represents all possible values for the length of AC, in feet?

1) 2 < AC < 8
2) 2 < AC < 8
3) 3 < AC < 7
4) 3 < AC < 7

Answer: 2) 2 < AC < 8

The Triangle Inequality Theorem states that the sum of the two shorter sides of a triangle must be greater than the length of the longest side.

You cannot have a triangle with side 2-3-5 or 3-5-8. Those would be straight lines. However, if the shortest side was even slightly greater than 2 or slightly less than 8, that would be okay. This is Choice (2).

Choice (3) is no good because it doesn't take into account any of the values between 2 and 3 or 7 and 8. Triangle lengths don't have to be integers.

More to come. Comments and questions welcome.

More Regents problems.

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