Geometry Problems of the Day (Geometry Regents, January 2013)

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, June 2013

Part I: Each correct answer will receive 2 credits.


26. The measure of an interior angle of a regular polygon is 120°. How many sides does the polygon have?

1) 5
2) 6
3) 3
4) 4

Answer: 2) 6

An equilateral triangle has 180° and each angle is 60°.

A square has 360° and each angle is 90°.

A regular pentagon has 540° and each angle is 108°.

A regular hexagon has 720° and each angle is 120°.

22. As shown in the diagram of rectangle ABCD below, diagonals AC and BD intersect at E.

If AE x  2 and BD 4x 16, then the length of AC is

1) 6
2) 10
3) 12
4) 24

Answer: 4) 24

The diagonal of a rectangle are congruent, so AC = BD, and AC = 2 * AE.

So

4x - 16 = 2(x + 2)
4x - 16 = 2x + 4
2x = 20
x = 10

AC = 2(x + 2) = 2(10 + 2) = 2(12) = 24

28. If the vertices of triangle ABC are A(-2,4), B(-2,8), and C(-5,6), then ABC is classified as

1) right
2) scalene
3) isosceles
4) equilateral

Answer: 3) isosceles

Immediately eliminate Choice (1) right because every right angle MUST also be either scalene, isosceles, so it cannot be the correct answer.

Use the distance formula or the Pytagorean Theorem to find the length of the sides. (You don't need to for AB.)

AB: SQRT( (-2 - -2)² + (8 - 4)² ) = SQRT(0 + 16) = 4

BC: SQRT( (-5 - -2)² + (6 - 8)² ) = SQRT(9 + 4) = SQRT(13)

CA: SQRT( (-2 - -5)² + (4 - 6)² ) = SQRT(9 + 4) = SQRT(13)

The triangle is isosceles.

End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

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