Geometry Problems of the Day (Geometry Regents, August 2010)

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 2010

Part I: Each correct answer will receive 2 credits.


25. In the diagram below of circle C, mQT = 140 and m∠P = 40

What is mRS ?

1) 50
2) 60
3) 90
4)100

Answer: 2) 60

Angle P will be one-half the difference of arc QT and RS.

1/2(140 - x) = 40

140 - x = 80

-x = -60

x = 60

Choice (2) is correct.

If you forget the 1/2, you would have gotten (4) 100 by mistake.

26. Which statement is logically equivalent to “If it is warm, then I go swimming”?

1) If I go swimming, then it is warm
2) If it is warm, then I do not go swimming.
3) If I do not go swimming, then it is not warm
4) If it is not warm, then I do not go swimming.

Answer: 3) If I do not go swimming, then it is not warm

The contrapositive is logically equivalent to the conditional statement.

The contrapositive negates both clauses and switches the order. That is, p -> q has the same truth values as ~q -> ~p.

Choice (1) is the converse. It can be warm without my going swimming.

Choice (2) is not the inverse because only one side is negated. It couldn't possibly have the same truth values.

Choice (4) is the inverse. It is possible that I will go swimming if it is not warm.

27. In the diagram below of △ACT, BE || AT.

If CB = 3, CA = 10, and CE = 6, what is the length of ET?

1) 5
2) 14
3) 20
4) 26

Answer: 2) 14

Read the question twice. What you need to compare is not what the question is asking for. Since the triangles are similar, the corresponding sides are proportional.

CB / CA = CE / CT

3 / 10 = 6 / CT

3 CT = 60

CT = 20

However, you are not asked for CT. You want ET. ET = CT - CE = 20 - 6 = 14.

28. Which geometric principle is used in the construction shown below?

1) The intersection of the angle bisectors of a triangle is the center of the inscribed circle.
2) The intersection of the angle bisectors of a triangle is the center of the circumscribed circle.
3) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the inscribed circle.
4) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the circumscribed circle.

Answer: 1) The intersection of the angle bisectors of a triangle is the center of the inscribed circle.

First of all, the circle is inscribed (inside), not circumscribed (outside). Eliminate Choice (2) and (4).

Next, the marks show bisecting an angle. However, nothing about those marks shows anything to be perpendicular. So the correct Choice is (1).

End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

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