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.


11. A plane intersects a cylinder perpendicular to its bases.

This cross section can be described as a

(1) rectangle
(2) parabola
(3) triangle
(4) circle

Answer: (1) rectangle

If the plane intersects perpendicular to the bases, then it is slicing through vertically. If you slice it vertically, you will get a rectangle.

You would get a circle if you cut horizontally.

If you cut at an angle you would get either an ellipse or a parabola.

You cannot get a triangle with a single cut.

12. An equation of line p is y = 1/3 x + 4. An equation of line q is y = 2/3 x + 8. Which statement about lines p and q is true?

(1) A dilation of 1/2 centered at the origin will map line q onto line p.
(2) A dilation of 2 centered at the origin will map line p onto line q.
(3) Line q is not the image of line p after a dilation because the lines are not parallel.
(4) Line q is not the image of line p after a dilation because the lines do not pass through the origin.

Answer: (3) Line q is not the image of line p after a dilation because the lines are not parallel.

A dilation of a line doesn't chage its orientation, which means that its slope will be unchanged.

The slope of line p is 1/3 and the slope of line q is 2/3 . These lines are not parallel, so one cannot be the image of the other after a dilation. This is Choice (3).

Neither line passes through the origin, but that has nothing to do with lines being a dilation or not.

13. The coordinates of the endpoints of SC are S(-7,3) and C(2,-6). If point M is on SC, what are the coordinates of M such that SM:MC is 1:2?

(1) (-4,0)
(2) (0,4)
(3) (-1,-3)
(4) (-5/2, -3/2)

Answer: (1) (-4,0)

Add 1 + 2 = 3. Point M is 1/3 of the way from S to C.

The distance from S to C is 2 - -7 = 9 and -6 - 3 = -9. One-third of 9 is 3 and onet-third of -9 is -3.

Point C is (-7 + 3, 3 - 3) = (-4, 0), which is Choice (1).

Choice (3) is at 2:1, or 2/3 of the way.

Choice (4) is the midpoint of SC, which is a 1:1 ratio.

14. On the set of axes below, rectangle WIND has vertices with coordinates W(-4,2), I(4,0), N(3,-4), and D(-5,-2).

What is the area of rectangle WIND?

(1) 17
(2) 31
(3) 32
(4) 34

Answer: (4) 34

Because it's a rectangle, you can find the lengths of DW and WI and multiply them. You can find the lengths using the distance formula or Pythagorean Theorem (which are the same thing!). Leave your answers in radical form.

DW = SQRT(4² + 1² = SQRT(17).

WI = SQRT(2² + 8² = SQRT(68).

SQRT(17) * SQRT(68) = SQRT(1156) = 34

or SQRT(17 * 17 * 2 * 2) = 17 * 2 = 34

Choice (4) is the answer.

15. In parallelogram ABCD shown below, EB bisects ∠ABC.

If m∠A = 40°, then m∠BED is

(1) 40°
(2) 70°
(3) 110°
(4) 140°

Answer: (3) 110°

If m∠A = 40°, then ∠C = 40° and m∠ABC = 140°.

If m∠ABC is bisected, then ∠ABE = 70° and m∠CBE = 70°.

Since ∠C = 40° and m∠CBE = 140° then angle BED, which is exterior to triangle BEC, must be 70 + 40 = 110°. This is Choice (3).

More to come. Comments and questions welcome.

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