## Sunday, July 07, 2024

### Geometry Problems of the Day (Geometry Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

### Part I

17. If ABCD is a parallelogram, which additional information is sufficient to prove that ABCD is a rectangle?

(1) AB ≅ BC
(2) AB || CD
(3) AC ≅ BD
(4) AC ⊥ BD

A rectangle is a parallelogram with right angles. One additional property of a rectangle is that its diagonals are congruent, but they are not necessary perpendicular.

Choice (1) makes the parallelogram a rhombus, not a rectangle. Eliminate Choice (1).

Choice (2) is true of every parallelogram. Eliminate Choice (2).

Choice (3) is true of rectangles. This is the correct answer.

Choice (4) is there to trick you. The sides being perpendicular, AB ⊥ BC, etc, are true of rectangles. However, the diagonals are only perpendicular if it is a rhombus (including squares). Eliminate Choice (4).

18. Line segment APB has endpoints A(-5,4) and B(7,-4). What are the coordinates of P if AP:PB is in the ratio 1:3?

(1) (-2,2)
(2) (-1,1.3)
(3) (1,0)
(4) (4,-2)

A ratio of 1:3 means that the line segment can be broken up into four equal (congruent) segments. AP has the legnth of one of those segments and PB has the length of three of those segments.

Because they used the ratio 1:3, point P is 1/4 of the way from A to B. That means, for this one problem, you can use a shortcut. (The "real" way is below.) You can find the midpoint of AB, call it M. Then find the midpoint of AM, which will be point P because 1/2 of 1/2 is 1/4.

M = ( (-5+7)/2, (4+(-4))/2 ) = (1,0)

P = ( (-5+1)/2, (4+0/2) ) = (-2,2), which is Choice (1).

The "right" way, and the way you would have to do it if the numbers weren't so nice is as follows:

x: 7 - (-5) = 12, and 1/4 (12) = +3.

y: -4 - 4 = -8, and 1/4 (-8) = -2

P = (-5 + 3, 4 - 2) = (-2, 2), which is Choice (1).

19. In the diagram below, AB and CD intersect at E, and CA and DB are drawn.

If CA || BD, which statement is always true?

(1) AE ≅ BE
(2) AE ≅ BE
(3) △AEC ~ △BED
(4) △AEC ≅ △BED

While the triangles are drawn to look like they are congruent, there is not enough information to same that they are congruent.

However, there are vertical angles (which are congruent) at point E, and because CA || BD, there are alternate interior angles that are congruent. That makes the triangles similar by AA Theorem. Choice (3) is correct.

Note that if the triangles were congruent, then they would also be similar, so two choices would be correct because it doesn't say "similar but not congruent".

Also, note that if Choice (1) was true, then Choice (3) would be true by ASA, which would make Choice (2) true as well by CPCTC.

You know that all four choices cannot be true.

20. If sin(3x + 9)° = cos(5x - 7)°, what is the value of x?

(1) 8
(2) 11
(3) 33
(4) 42

The following is always true: sin (x) = cos (90 - x).

Therefore,

3x + 9 = 90 - (5x - 7)
3x + 9 + 5x - 7 = 90
8x + 2 = 90
8x = 88
x = 11

Checking out work: 3(11) + 9 = 42, 5(11) - 7 = 48, and 42 + 48 = 90

Choice (2) is the correct answer.

21. Which set of integers could represent the lengths of the sides of an isosceles triangle?

(1) {1,1,3}
(2) {2,2,5}
(3) {3,3,6}
(4) {4,4,7}

For a triangle to be isosceles, two of the sides must be the same length. This is true of all four choices.

However, for three lines to make a triangle, the sum of the lengths of the two shorter sides must be longer than the length of the longest side. Longer, not equal.

Choices (1), (2), and (3) do not make triangles. The sides are too short. Only Choice (4) makes a triangle, so it is the correct answer.

22. In the diagram shown below, altitude CD is drawn to the hypotenuse of right triangle ABC.

Which equation can always be used to find the length of AC?

(2) CD/AC = AC/AB
(3) AC/CD = CD/BC

An altitude drawn from a right angle to the hypotenuse of a right triangle will divide the right triangle is such a way that the two smaller triangles will be similar to each other and to the original triangle. This is the Right Triangle Altitude Theorem.

This means that the corresponding sides of the triangles are proportional.

Choice (1) only uses sides from one triangle. It doesn't use corresponding pairs of sides. Eliminate Choice (1).

In Choice (2), CD doesn't correspond to AC in triangle ABC, so the proportion is no good. Eliminate Choice (2).

In Choice (3), the proportion says hypotenuse / short leg = long leg / hypotenuse. Elimnate Choice (3).

Choice (4) has hypotenuse / long leg = hypotenuse / long leg. This is the correct answer.

23. Which congruence statement is sufficient to prove parallelogram MARK is a rhombus?

(1) MA ≅ MK
(2) MA ≅ KR
(3) ∠K ≅ ∠A
(4) ∠R ≅ ∠A

To show that a parallelogram is also a rhombus, you need to show that either the consecutive sides are congruent or that they diagonals are perpendicular. (The diagonals do not have to be congruent, and will only be congruent in a square or rectangle.)

There are no choices suggesting that the diagonals are perpendicular, so that means congruent sides.

Choice (1) has consecutive sides MA and MK. This is the correct answer.

Choices (2) and (3) are true for any parallelogram.

If Choice (4) is true in a parallelogram, then the parallegram is a rectangle because the angles would both have to be 90 degrees because the sum of angles A and R is 180.

24. A line whose equation is y = -2x + 3 is dilated by a scale factor of 4 centered at (0,3). Which equation represents the image of the line after the dilation?

(1) y = -2x + 3
(2) y = -2x + 12
(3) y = -8x + 3
(4) y = -8x + 12

Answer: (1) y = -2x + 3

Dilating a line will NOT change its direction which means that the slope DOES NOT change. Eliminate Choices (3) and (4) immediately.

When you dilate a line, you either get a line that is parallel to the original line OR you get the original line if the center of dilation was a point on that line.

Since (0,3) is a point on the line y = -2x + 3 (Proof: 3 = -2(0) + 3), the image will be the same line as the pre-image. So Choice (1) is the correct answer.

End of Part I.

How did you do?

More Regents problems.

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