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

Part I: Each correct answer will receive 2 credits.


16. The diagram below shows a pair of congruent triangles, with ∠ADB ≅ ∠CDB and ∠ABD ≅ ∠CBD.

Which statement must be true?

1) ∠ADB ≅ ∠CBD
2) ∠ADB ≅ ∠CBD
3) AB ≅ CD
4) AD ≅ CD

Answer: 4) AD ≅ CD

By ASA the triangles are congruent, so the corresponding parts are also congruent.

In Choice (1) ∠ADB does not correspond to ∠CBD.

In Choice (2) ∠ABC and ∠ABC are not related in any way. They are opposing angles of a kite, and we have no information about them.

In Choice (3) AB does not correspond to CD.

In Choice (4) AD does correspond to CD.

17. What is an equation of the line that is perpendicular to the line whose equation is y = 5/3 x - 2 and that passes through the point (3,-6)?

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

Answer: 3) y = -5/3 x - 1

A line perpendicular to a line with a slope of 3/5 will have a slope of -5/3, the inverse reciprocal. The product of (3/5)(-5/3) = -1. Eliminate Choices (1) and (4).

Substitute 3 for x and see which equation gives you -6.

y = -5/3 (3) + 11 = -5 + 11 = 6. Eliminate Choice (2).

y = -5/3 (3) - 1 = -5 - 1 = -6. Choice (3) is the answer.

18. Point A lies in plane B. How many lines can be drawn perpendicular to plane B through point A?

1) one
2) two
3) zero
4) infinite

Answer: 1) one

Think about sticking a pencil through a hole in a sheet of looseleaf. There is only one way to hold it so that it is perpendicular to the plane through that point.To negate a statement, you have to add a "not", or remove a "not" that is already there.

The Leaning Tower of Pisa is not perpendicular to the ground. Of course, if it were, it wouldn't be as famous. If they wanted to have a perpendicular tower, there is only one line it could have.

19. In the diagram below of isosceles trapezoid ABCD, AB = CD = 25, AD = 26, and BC = 12.

What is the length of an altitude of the trapezoid?

1) 7
2) 14
3) 19
4) 24

Answer: 4) 24

If you know you're Pythagorean Triples, you would recognize that this was going to be a Pythagorean Theorem problem, and the two most common right triangles with a hypotenuse of 25 are the 15-20-25 and 7-24-25 triangles. Just sayin'.

Otherwise...

The trapezoid is isosceles, so it altitudes were drawn from AD to point B and to point D, the two right triangles would be congruent by Hypotenuse-Leg. The base of those triangles could be found using the equation:

26 - 2x = 12
14 = 2x
x = 7 This means that the base of the right triangle is 7, and its hypotenuse is 25. Therefore 7² + x² = 25²
49 + x² = 625
x² = 576
x = 24

Knowing Pythagorean Triples can save you a bit of work with multiple-choice problems.

20. What is an equation of circle O shown in the graph below?

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

Answer: 1) (x + 2)² + (y - 2)² = 9

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.

Since the radius is 3, then r² = 9. Eliminate Choices (2) and (4).

Since the signs are flipped, the correct answer is Choice (1).

More to come. Comments and questions welcome.

More Regents problems.

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