Geometry Problems of the Day (Geometry Regents, January 2013)

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.


21. In the diagram of trapezoid ABCD below, AB || DC, AD ≅ BC, m∠A = 4x + 20, and m∠C = 3x - 15.

What is m∠D?

1) 25
2) 35
3) 60
4) 90

Answer: 3) 60

There is no relation between angles A and C unless you know that the trapezoid is isosceles. You know this is the case because AD ≅ BC.

You know that AB || CD, and therefore the consecutive angles A and D are interior angles on the same side of a transversal. That makes them supplementary. Since the trapezoid is isosceles, the base angles are congruent.

So 4x + 20 + 3x - 15 = 180
7x + 5 = 180
7x = 175
x = 25

Angle D is congruent to angle C, which is 3(25) - 15 = 75 - 15 = 60.

22. In circle R shown below, diameter DE is perpendicular to chord ST at point L

Which statement is not always true?

1) SL ≅ TL
2) RS = DR
3) RL ≅ LE
4) (DL)(LE) = (SL)(LT)

Answer: 3) RL ≅ LE

When a diameter is perpendicular to a chord, it will bisect the chord. Also, when any two chords intersect (including a diameter) the product of the lengths of the two segments of the chords will be equal.

Choice (1) is always true because ST is bisected.

Choice (2) is always true because both RS and DR are radii. All radii in a circle are equal in length.

Choice (3) is NOT always true. Most of the time it will NOT be true. The chord does not have to bisect the radius. It could but doesn't have to.

Choice (4) is always true for any two chords that intersect, perpendicular or not.

23. Which equation represents circle A shown in the diagram below?

1) (x - 4)² + (y - 1)² = 3
2) (x + 4)² + (y + 1)² = 3
3) (x - 4)² + (y - 1)² = 9
4) (x + 4)² + (y + 1)² = 9

Answer: 4) (x + 4)² + (y + 1)² = 9

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

It should be obvious that the radius is 3 and 3² = 9, so eliminate choices (1) and (2).

The signs for h and k are flipped because of the minus signs, and the center is (-4, -1) so h = 4 and k = 1 NOT -4 and -1, respectively.

Note that there was another circle equation problem Five Questions ago, in yesterday's post.

24. Which equation represents a line that is parallel to the line whose equation is 3x - 2y = 7?

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

Answer: 3) y = 3/2 x - 5

Parallel lines have the same slope, so we need to know the slope of the given line. The choices are all in slope-intercept form.

The slope of a line in Standard Form is -A/B, or -3/-2 = 3/2.

Alternatively, rewrite 3x - 2y = 7
-2y = -3x + 7
y = 3/2x - 7/2

The slope of the two parallel lines must be 3/2.

25. In the diagram below of circle O, PAC and PBD are secants.

If mCB = 70 and mAB = 20, what is the degree measure of ∠P?

1) 25
2) 35
3) 45
4) 50

Answer: 1) 25

The formula for angle P is (CD - AB) / 2.

(70 - 20) / 2 = 50 / 2 = 25

More to come. Comments and questions welcome.

More Regents problems.

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