Geometry Problems of the Day (Geometry Regents, August 2010)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, August 2010

Part I: Each correct answer will receive 2 credits.


11. In △ABC, AB = 7, BC = 8, and AC = 9. Which list has the angles of △ABC in order from smallest to largest?

1) ∠A, ∠B, ∠C
2) ∠B, ∠A, ∠C
3) ∠C, ∠B, ∠A
4) ∠C, ∠A, ∠B

Answer: 4) ∠C, ∠A, ∠B

The sides from smallest to largest are AB = 7, BC = 8, and AC = 9.

Therefore, the angles from smallest to largest are ∠C, ∠A, ∠B, which is Choice (4).

12. Tangents PA and PB are drawn to circle O from an external point, P, and radii OA and OB are drawn. If m∠APB = 40, what is the measure of ∠AOB?

1) 140°
2) 100°
3) 70°
4) 50°

Answer: 1) 140°

∠AOB is supplementary to ∠APB. The two tangents and two radii form a quadrilateral (more specifically, a kite), which has 360°. The tangents meet the radii at right angles, and 90 + 90 = 180.

So 180 - 40 = 140.

13. What is the length of the line segment with endpoints A(−6,4) and B(2,−5)?

1) √13
2) √17
3) √72
4) √145

Answer: 4) √145

Pythagorean Theorem or Distance Formula.

The difference in the x values is 8. The difference in the y values is 9.

√(8² + 9²) = √(64 + 81) = √145, which is Choice (4).

14. The lines represented by the equations y + 1/2 x = 4 and 3x + 6y = 12 are

1) the same line
2) parallel
3) perpendicular
4) neither parallel nor perpendicular

Answer: 2) parallel

Rewrite both equations in the same form, such as slope-intercept.

y + 1/2 x = 4
y = -1/2 x + 4

3x + 6y = 12
6y = -3x + 12
y = -1/2x + 2

The two lines have the same slope but different y-intercepts, so the lines are parallel.

15. A transformation of a polygon that always preserves both length and orientation is

1) dilation
2) translation
3) line reflection
4) glide reflection

Answer: 2) translation

A translation preserves length and orientation.

A dilation does not preserve length.

A reflection (line or glide) does not preserve orientation.

More to come. Comments and questions welcome.

More Regents problems.

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