## Monday, November 08, 2021

### Geometry Problems of the Day (Geometry Regents, January 2012)

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, January 2012

21. The diagram below represents a rectangular solid.

Which statement must be true?

1) EH and BC are coplanar.
2) FG and AB are coplanar.
3) EH and AD are skew.
4) FG and CG are skew.

Answer: 1) EH and BC are coplanar.

Coplanar lines share a plane. Both parallel lines and intersecting lines will share a plane. If they were pencils, you could lay a sheet a paper on top of them (gravity notwithstanding). Two lines are skew if they don't lie in the same plane, like a hallway and an elevator shaft. Skew lines pass by each other but don't touch.

In Choice (1), both EH and BC are vertical lines. They share a plane. This is the answer.

In Choice (2), FG is vertical but AB is horizontal (left to right). These lines do not share a plane.

In Choice (3), EH and AD are both vertical lines. They share a plane, so they are NOT skew.

In Choice (4), FG and CG intersect at point G (duh!), so they share a plane.

22. In triangle RST, m∠R = 58 and m∠S = 73. Which inequality is true?

1) RT < TS < RS
2) RS < RT < TS
3) RT < RS < TS
4) RS < TS < RT

Answer: 4) RS < TS < RT

In a triangle, the shortest side will be across from the smallest angle. The longest side will be across from the biggest angle. In triangle RST, ST will be across from R, TR will be across from S, and RS will be across from T.

Since angle R is smaller than angle S, then side TS < side RT. This eliminates Choices (1), (2) and (3). The answer must be Choice (4).

There are 180 degrees in the triangle, and you were given two of the three angles. 180 - 58 - 73 = 49. So angle T is the smallest angle and side RS is the shortest side. So RS < TS < RT.

23. The number of degrees in the sum of the interior angles of a pentagon is

1) 72
2) 360
3) 540
4) 720

If you draw the diagonals from one point of a pentagon you can create three triangles, each of which has 180 degrees inside of it.

Therefore, a pentagon has 180 * 3 = 540 degrees, which is Choice (3).

Choice (1) makes no sense because it is less than 180 degrees, which makes one triangle. It's not a totally bizarre answer though because it is ... are you ready for this ... the size of a single exterior angle in a regular pentagon, just in case you really, really misread the problem.

Choice (2) is a quadrilateral of any kind. There are also 360 degrees in a circle, but that isn't what they were asking about.

Choice (4) is a hexagon, which can be divided into four triangles.

Basically, the formula is (n - 2)(180 degrees). The "- 2" is because no matter how many sides the polygon has, you can divide it up into two fewer triangles.

24. The coordinates of the endpoints of AB are A(0,0) and B(0,6). The equation of the perpendicular bisector of AB is

1) y = -1/2 x
2) y = -1/2 x + 1
3) y = 2x - 5
4) y = 2x - 1

Answer: 3) y = 2x - 5

A parallel line has the same slope, so eliminate Choices (1) and (2) immediately.

Plus in x = 2 and see which equation gives you -1.

y = 2(2) - 5 = -1. This is the correct choice.

y = 2(2) - 1 = 3. Eliminate Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

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