## Saturday, July 06, 2024

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

This exam was adminstered in June 2024.

More Regents problems.

### Part I

9. In circle O below, chords CT and EN intersect at point A. Chords CB and NT are drawn.

Which statement is always true?

(1) NT/TA = CB/BA
(2) ∠BAC ≅ ∠ATN
(3) NA/AB = TA/AC
(4) ∠BCA ≅ ∠NTA

Once again the triangles are similar (not congruent). If they weren't congruent, then there would be no answer to this question. This means that the corresponding angles are congruent and the corresponding sides are proportional.

You can see that the triangles are similar because there are vertical angles at point A which must be congruent, and angle C and angle N are both inscribed angles that intercept the same arc, BT, so they must be congruent. By AA, the triangles are similar.

Choice (1) says NT/TA = CB/BA. These are two pairs of corresponding sides in the correct order. This statement is true. Choice (1) is the correct answer.

In Choice (2), a vertical angle is being compared to an inscribed angle. There is no reason for these to be congruent. Eliminate Choice (2).

In Choice (3), the sides are not listed correctly. In fact, if there was multiplication instead of division, the statement would've been correct. Eliminate Choice (3).

Choice (4) compares angles C with angle T. As we have alreday said, angle C is congruent to angle N, not T. Angle T is congruent to angle B because both are inscribed angles that mark off arc CN. Eliminate Choice (4).

10. In the diagram below of △ABC, CBF is drawn, AB bisects ∠FBD, and BD ⟂ AC.

If m∠C = 42o, what is m∠A?

(1) 24o
(2) 33o
(3) 48o
(4) 66o

Follow along on the diagram below. You have to hit most of the angles in the diagram to get the answer.

Angle C is 42 degrees. Angle CDB is 90 degrees because BD is perpendicular to AC. That means that angle CBD is 48 degrees because CBD is a triangle and the angles must total 180 degrees.

Angles CBD, DBA, and ABF form a straight line and must add up to 180 degrees. Because AB bisects FBD, then angles DBA and ABF must be equal in measure, so we can label them both x.

That means 42 + x + x = 180
42 + 2x = 180
2x = 132
x = 66

If angle DBA = 66 and angle BDA = 90, then angle A = 180 - 90 - 66 = 24 degrees, which is Choice (1).

11. In circle O below, OA = 6, and m∠COA = 100°.
What is the area of the shaded sector?

(1) 10 π
(2) 26 π
(3) 10 π / 3
(4) 26 π / 3

The area of the circle is A = π r2. The area of the shaded sector is the area of the circle times the fraction of the circle that's shaded in.

The central angle is 100°, and that sector is NOT shaded in. That means that the shaded portion is 260°/360° or 260/360, which simplifies to 13/18. (If you're using a calculator, you don't need to simplify).

Leave the answer in terms of pi (That is, don't hit the pi key on the calculator or enter a number for it) because all the answers have pi in them.

A = 260/360 (6)2 = 26

The correct answer is Choice (2).

12. In rectangle ABCD, diagonal AC is drawn. The measure of ∠ACD is 37° and the length of BC is 7.6 cm. What is the length of AC, to the nearest tenth of a centimeter?

(1) 4.6
(2) 6.1
(3) 10.1
(4) 12.3

If you sketch a little diagram, you will see that 7.6 is opposite the 37 degree angle and that the diagonal is the hypotenuse of a right triangle.

Opposite and hypotenuse means that you need to use the sine ratio.

sin 37 = 7.6 / x
x = 7.6 / sin 37 = 12.62...

The correct answer is Choice (4).

Note that if you use cosine or tangent, you will likely get one of the other choices.

Also, since the hypotenuse MUST BE longer than 7.6, you could immediately eliminate Choices (1) and (2). They are too short.

13. A peanut butter manufacturer would like to use a cylindrical jar with a volume of 1180 cm3. The jar has a height of 10 cm. What is the diameter of the jar, to the nearest tenth of a centimeter?

(1) 3.8
(2) 6.1
(3) 10.9
(4) 12.3

Notice that they asked for the diameter and not the radius. The formula uses radius, and that is you must solve for first. Then you can get the diameter.

The formula for volume of a cylinder is V = π r2 h. Plug in what you know to find out what you don't know.

1180 = (3.141592)(r)2(10)
r2 = 1180 / ((3.141592)(10))
r = √(1180 / ((3.141592)(10)) = 6.128...

So the diameter is 6.128 * 2 = 12.257... or 12.3, which is Choice (4).

14. Triangle KLM is dilated by a scale factor of 3 to map onto triangle DRS. Which statement is not always true?

(1) ∠K ≅ ∠D
(2) KM = 1/3 DS
(3) The area of triangle DRS is 3 times the area of triangle KLM.
(4) The perimeter of triangle DRS is 3 times the area of triangle KLM.

Answer: (3) The area of triangle DRS is 3 times the area of triangle KLM.

When you dilatte a polygon, length (such as perimeter) will be increased by the scale factor (multiplied by the scale factor). However, the area will be the original area times the square of the scale factor, because both the length and the width have increased.

The correct choice is Choice (3).

The corresponding angles will remain congruent because the two triangles are similar. And the legnth of side DS will be 3 times bigger than side KM, so KM is 1/3 DS.

15. A rectangle with dimensions of 4 feet by 7 feet is continuously rotated about one of its 4-foot sides. The resulting three-dimensional object is a

(1) cylinder with a height of 7 feet and a base radius of 4 feet.
(2) cylinder with a height of 4 feet and a base radius of 7 feet.
(3) cone with a height of 7 feet and a base radius of 7 feet.
(4) cone with a height of 4 feet and a base radius of 7 feet.

Answer: (2) cylinder with a height of 4 feet and a base radius of 7 feet.

If a rectangle is rotated about an axis, there is no way that they resulting shape could be a cone. It will be a cylinder. Eliminate Choices (3) and (4).

If it is rotated about the 4-foot side then 4 feet will be the height of the cylinder. The rectangle will swing in a circle with a radius of 7 feet.

The correct answer is Choice (2).

16. In right triangle ABC, altitude CD is drawn to hypotenuse AB. If AD = 4 and CD = 8, the length of BD is

(1) √(48)
(2) √(80)
(3) 12
(4) 16

The Right Triangle Altitude Theorem tells us that (CD)2 = (AD)(BD). Substitute and solve.

(8)2 = (4)(x)
64 = 4x
x = 16

The correct answer is Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

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