Geometry Problems of the Day (Geometry Regents, August 2022)

This exam was adminstered in August 2022.

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August 2022 Geometry Regents

Part I

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


16. In right triangles ABC and RST, hypotenuse AB = 4 and hypotenuse RS = 16. If △ABC ∼ △RST, then 1:16 is the ratio of the corresponding

(1) legs
(2) areas
(3) volumes
(4) perimeters

Answer: (2) areas

If the two hypotenuses are 4 and 16, then the ratios of those lengths is 1:4.

The legs and the perimeters also have ratios of 1:4.

The areas have a ratio of 1:4² or 1:16.

The volumes don't make sense because they are triangles and not prisms. However, if they were prisms, then the ratios of the volumes would have been 1:4³ or 1:64.

The correct answer is Choice (2).

17. Parallelogram ABCD with diagonal DB is drawn below. Line segment EF is drawn such that it bisects DB at M.

Which triangle congruence method would prove that △EMB ≅ △FMD?

(1) ASA, only
(2) AAS, only
(3) both ASA and AAS
(4) neither ASA nor AAS

Answer: (3) both ASA and AAS

Lines AB and CD are parallel. These lines are crossed by two transversals, EF and DB, and each of these creates alternate interior angles. The two transversal intersect, creating vertical angles. So anyone could pick any two angles they wanted to to prove congruence.

Since DB is bisected at M, the DM = MB. Depending upon which angles where used for your proof, you may have either ASA, if the side is included, or AAS, if the side is not included between the two pairs of angles.

The correct answer is Choice (3).

18. In the diagram below of circle O, chords AD and BC intersect at E, and chords AB and CD are drawn.

Which statement must always be true?

(1) AB ≅ CD
(2) AD ≅ CD
(3) ∠B ≅ ∠C
(4) ∠A ≅ ∠C

Answer: (4) ∠A ≅ ∠C

Angles A and C are both inscribed angles and they intercept the same arc BD. Therefore both angles, A and C, must have a measure of 1/2 the size of arc BD.

There is no reason for chords AB nd CD to be congruent. They could be if they are equidistant from point O, but there is no information that they are.

Similarly, there is no reason for AD and BC to be congruent. There are rules about there products of their parts, but not about the chords themselves.

For angles B and C, they would be congruent only if AB || CD. There is nothing to suggest that those chords are parallel.

The correct choice is Choice (4).

19.What are the coordinates of the center and length of the radius of the circle whose equation is x² + y² - 12y - 20.25 = 0?

(1) center (0,6) and radius 7.5
(2) center (0,-6) and radius 7.5
(3) center (0,12) and radius 4.5
(4) center (0,-12) and radius 4.5

Answer: v

The formula for 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.

To find h,k, and r, you need to Complete the Square to rewrite the equation in the proper form.

HOWEVER, look at the four choices. There are four different y co-ordinates, meaning that once you have that, you know the correct choice.

To complete the square, you need to halve the coefficent of the y term. Half of -12 is -6. However, the formula have a minus sign in it. So if (y - 6)² is in the equation, then the center is (0,6), which is Choice (1).

To prove it:

x² + y² - 12y - 20.25 = 0
x² + y² - 12y = 20.25
x² + y² - 12y + 36 = 20.25 + 36
x² + (y - 6)² = 56.25
x² + (y - 6)² = 7.5²

The center is (0,-6) and the radius is 7.5.

20. In the diagram below, ABCD is a rectangle, and diagonal BD is drawn. Line ℓ, a vertical line of symmetry, and line m, a horizontal line of symmetry, intersect at point E.
Which sequence of transformations will map △ABD onto △CDB?

(1) a reflection over line ℓ followed by a 180° rotation about point E
(2) a reflection over line ℓ followed by a reflection over line m
(3) a 180° rotation about point B
(4) a reflection over DB

Answer: (2) a reflection over line ℓ followed by a reflection over line m

A reflection over line ℓ followed by a reflection over line m will move A to B then to C, move B to A then to D, and move D to C then to B, so ABD goes to CDB. This is Choice (2).

In Choice (1), a rotation of 180 degrees about point E would by itself map ABD onto CDB. However, the reflection of line l first messes this up. Visualize it like this: the reflection will change the diagonal from a positive slope to a negative slope, and a rotation will leave it with a negative slope (from upper left to lower right).

In Choice (3), rotating about point B will put the rectangle somewhere in the upper right of the graph. It won't map it onto itself.

In Choice (4), a reflection of DB would change the orientation of the rectangle, so it would be impossible for one triangle to map onto the other.

More to come. Comments and questions welcome.

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