Wednesday, January 25, 2023

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.

1. In the diagram below, △ABC is reflected over line ℓ to create △DEF.

If m∠A = 40° and m∠B = 95°, what is m∠F?

(1) 40°
(2) 45°
(3) 85°
(4) 95°

Answer: (2) 45°

In this reflection A maps to D, B maps to E, and C maps to F. So the measure of angle F is the same as the measure of angle C.

If A measures 40 and B measure 95, and 40 + 95 = 135, then the measure of angle C is 180 - 135 = 45 degrees, which is Choice (2).

2. The diagram below shows triangle ABC with point X on side AB and point Y on side CB.

Which information is sufficient to prove that △BXY ∼ △BAC?

(1) ∠B is a right angle.
(2) XY is parallel to AC
(3) triangle ABC is isosceles.
(4) AX ≅ CY

Answer: (2) XY is parallel to AC

Similarity is proven is two pairs of corresponding angles are congruent. This makes the third pair congruent as well. Having the same size angles means the triangle has the same shape. Congruent sides are not neceassry for similarity. In fact, sides most likely will NOT be congruent because they it will not longer be a similarity problem but a congruency problem.

Angle B is congruent to itself by the Reflexive Property. It doesn't matter what the size of Angle B is. Eliminate Choice (1).

If the lines are parallel, then the corresponding angles are congruent. So if angle ACB is congruent angle XYB, that, along with angle B being congruent to itself, is enough to prove AA Similarity. CHoice (2) is the correct answer.

Choice (3) would be good IF we also knew that triangle BXY was also isosceles, because then we could write equal expressions for the sizes of all the angles, even if we didn't know the sizes of the angles themselves. However, only knowing that one triangle is isosceles is not sufficient.

In Choice (4), knowing that AX = CY doesn't help because we don't know the relationship between BY and BX to set up a proportion, and use SAS similarity instead of AA. Eliminate Choice (4).

3. Quadrilateral MATH is congruent to quadrilateral WXYZ. Which statement is always true?

(1) MA = XY
(2) m∠H = m∠W
(3) Quadrilateral WXYZ can be mapped onto quadrilateral MATH using a sequence of rigid motions.
(4) Quadrilateral MATH and quadrilater WXYZ are the same shape, but not necessarily the same size.

Answer: (3) Quadrilateral WXYZ can be mapped onto quadrilateral MATH using a sequence of rigid motions.

If they are congruent, then they are the came size, so Eliminate Choice (4).

Congruent polygons can be mapped onto each other using a series of rigid motions, such as translations, reflections or rotations. But no dilations. Choice (3) is the correct answer.

Choice (1) is not true. By defualt, MA = WX. I say by default because I have seen math teachers who are a little sloppy with this when naming their polygons (particularly with triangles). But the first letter maps to the first letter, the second to the second, etc., unless otherwise specified.

Choice (2) is incorrect as well for the same reasons as Choice (1).

4. A quadrilateral has diagonals that are perpendicular but not congruent. This quadrilateral could be

(1) a square
(2) a rhombus
(3) a rectangle
(4) an isocelese trapezoid

Answer: (2) a rhombus

The shape is a rhombus.

Squares have diagonals that are perpendicular and congruent, so this is incorrect.

Rhombuses (rhombi) have diagonals that are perpendicular. If they are also congruent, then the rhombus is also a square, but they do NOT have to be congruent.

A rectangle has diagonals that are congruent. If they are also perpendicular, then the rectangle will be a square.

An isosceles trapezoid has congruent diagonals.

5. Which regular polygon has a minimum rotation of 36° about it center that carries the polygon onto itself.

(1) pentagon
(2) octagon
(3) nonagon
(4) decagon

Answer: (4) decagon

For a regular polygon to carry onto itself, it must rotate a multiple of 360 degrees divided by the number of sides that the polygon has.

A pentagon rotates 360 / 5 = 72 degrees.

An octogon rotates 360 / 8 = 45 degrees.

A nonagon rotates 360 / 4 = 40 degrees.

A decagon rotates 360 / 10 = 36 degrees. CHoice (4) is correct.

More to come. Comments and questions welcome.

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