Geometry Problems of the Day (Geometry Regents, June 2023)

This exam was adminstered in June 2023.

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June 2023 Geometry Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


9. An equation of circle M is x² + y² + 6x - 2y + 1 = 0. What are the coordinates of the center and the length of the radius of circle M?

(1) center (3, -1) and radius 9
(2) center (3, -1) and radius 3
(3) center (-3, 1) and radius 9
(4) center (-3, 1) and radius 3

Answer: (4) center (-3, 1) and radius 3

You have to complete the squares for the x² term and the y² term. Thankfully, the choices give you hints.

Rewrite the equation grouping the x terms and the y terms and then move the constant to the right side of the equation.

x² + 6x + y² - 2y + 1 - 1 = 0 - 1

x² + 6x + y² - 2y = -1

To complete the squares, take have of 6, which is 3, and then square it, which is 9. Take half of -2, which is -1, and square it, which is 1. Add 9 + 1 to both sides of the equation.

x² + 6x + 9 + y² - 2y + 1 = -1 + 10

(x + 3)² + (y - 1)² = 9

The equation of a circle is given by the formula (x - h)² + (y - k)² = r², where (h,k) is the center of the circle and r is the radius.

So the center of the circle is (-3,1), because the signs flip to make 0, and the radius is 3, because r² is 9. Choice (4) is the correct answer.

10. Parallelogram BETH, with diagonals BT and HE, is drawn below.

Which additional statement is sufficient to prove that BETH is a rectangle?

(1) BT ⊥ HE
(2) BE || HT
(3) BT ≅ HE
(4) BE ≅ ET

Answer:(3) BT ≅ HE

If a parallelogram has diagonals that are congruent then it is a rectangle. Choice (3) is the correct answer.

Perpendicular diagonals make a parallelogram a rhombus, not a rectangle. Eliminate Choice (1).

Parallel sides are part of the definition of a parallelogram. Eliminate Choice (2).

Consecutive sides being congruent makes a parallelogram into a rhombus, not a rectangle. Eliminate Choice (4).

11. A gardener wants to buy enough mulch to cover a rectangular garden that is 3 feet by 10 feet. One bag contains 2 cubic feet of mulch and costs $3.66. How much will the minimum number of bags cost to cover the garden with mulch 3 inches deep?

(1) $3.66
(2) $10.98
(3) $14.64
(4) $29.28

Answer: (3) $14.64

Volume of a cone is V = L * W * H, but make sure that you are using the correct units, which must be the same.

V = 3 * 10 * .25 = 7.5. Since one bag covers 2 cubic feet, he needs 4 bags (because three isn't enough and they won't sell half a bag.)

The cost is then $3.66 * 4 = $14.64, which is Choice (3).

12. In the diagram below, &9651;DOG ~ &9651;CAT, where ∠G and ∠T are right angles.

Which expression is always equivalent to sin D?

(1) cos A
(2) sin A
(3) tan A
(4) cos C

Answer: (1) cos A

Sin D = sin C, but that isn't one of the choices.

However, the sine of one complementary angle is equal to the cosine of the other complementary angle in a right triangle.

That means that sin D = cos O = cos A. Cos A is Choice (1), which is the correct answer.

13. On the set of axes below, nDEF is the image of nABC after a dilation of scale factor 1/3.

The center of dilation is at

(1) (0,0)
(2) (2,-3)
(3) (0,-2)
(4) (-4,0)

Answer: (2) (2,-3)

Take a straightedge and draw line AD, extending it past D. Do the same for BE and CF.

The point of concurrence is (2,-3), which is the center of dilation. Choice (2) is the correct answer.

You can see that from (2,-3) to point F is a translation of -2,+1, and from (2,-3) to C is -6,+3, which is 3 times farther. Or F is only 1/3 of the distance to C.

You can check the distance to D and A and you will find that D is 1/3 of the way to A. Likewise, E is 1/3 of the way to B.

14. In the diagram below of isosceles triangle AHE with the vertex angle at H, CB ⊥ AE and FD ⊥ AE.
Which statement is always true?

(1) AH/AC = EH/EF
(2) AC/EF = AB/ED
(3) AB/ED = CB/FE
(4) AD/AB = BE/DE

Answer: (2) AC/EF = AB/ED

Because AHE is isosceles, angles A is congruent to angle E. Angle ABC and EDF are right angles, so they are congruent to each other. Therefore, by AA, triangles ABC and EDF are similar (not congruent), and that makes the corresponding sides proportional.

Choice (1) uses point H, comparing part of the side to the length of the entire side of triangle AHE. This would imply that AC ≅ EF, which isn't necessarily true. (It doesn't look it.) Eliminate Choice (1).

Choice (2) compares the ratio of the two hypotenuses to the ratio of the two bases. Choice (2) is the correct answer.

Choice (3) is incorrect because the first ratio compares the bases of the two triangles but the second ratio compares the vertical leg of one triangle to the hypotenuse of the other.

Choice (4) uses line segment BD, which is not part of either of the two similar triangles. Eliminate Choice (4).

15. Rectangle ABCD has two vertices at coordinates A(-1,-3) and B(6,5). The slope of BC is

(1) -7/8
(2) 7/8
(3) -8/7
(4) 8/7

Answer: (1) -7/8

Sides AB and BC must be perpendicular to each other, so the slopes of the two sides must be inverse reciprocals. That is, flip the fraction over, and switch the sign.

The slope of AB is (-3 - 5) / (-1 - 6) = -8/-7 = 8/7. That means that the negative reciprocal is -7/8, which is Choice (1).

16. In right triangle ABC, m∠A = 90^o, m∠B = 18^o, and AC = 8. To the nearest tenth, the length of BC is

(1) 2.5
(2) 8.4
(3) 24.6
(4) 25.9

Answer: (4) 25.9

AC is Opposite of angle B, and BC is the hypotenuse of the triangle. Opposite and hypotenuse means that we need to use the sine ratio.

sin 18 = 8 / x
so x = 8 / (sin 18) = 25.88...

This rounds to 25.9, so Choice (4) is the correct answer.

More to come. Comments and questions welcome.

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