Geometry Problems of the Day (Geometry Regents, June 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 III: Each correct answer will receive 4 credits. Partial credit is possible.


35. The coordinates of the vertices of parallelogram SWAN are S(2,-2), W(-2,-4), A(-4,6), and N(0,8). State and label the coordinates of parallelogram S"W"A"N", the image of SWAN after the transformation T_4,–2 ° D _1/2.
[The use of the set of axes below is optional.]

Answer:

When you have a Composition of Transformations, read the dot as "of the". You want to find the Translation OF THE Dilation. The order matters. If you do the transformations in the incorrect order, you will lose Half of the credit immediately.

You can sketch the points on a coordinate plane as a visual aid, but it isn't necessary. It can be done algebraically.

A dilation of 1/2, centered on the origin, means that all of the coordinates will be cut in half.

S(2,-2) --> S'(1,-1)
W(-2,-4) --> W'(-1,-2)
A(-4,6) --> A'(-2,3)
N(0,8) --> N'(0,4)

Next, add 4 to every x-coordinate and subtract 2 from every y-coordinate.

S(2,-2) --> S'(1,-1) --> S"(5, -3)
W(-2,-4) --> W'(-1,-2) --> W"(3, -4)
A(-4,6) --> A'(-2,3) --> A"(2, 1)
N(0,8) --> N'(0,4) --> N"(4, 2)

These all must be labeled.

36. In circle O shown below, chords AB and CD and radius OA are drawn, such that AB ≅ CD, OE ⊥ AB, OF ⊥ CD, OF = 16, CF = y + 10, and CD = 4y - 20.

Determine the length of DF.

Determine the length of OA.

Answer:

There is a lot going on in this image. Write down the things you know.

You need to find the length of DF, which you can do because DF ≅ CD, which is half the length of CD. You know this is true because OF is part of a radius which intersects CD at a right angle, which means that it bisects the chord.

So

4y - 20 = 2(y + 10)
4y - 20 = 2y + 20
2y = 40
y = 20

DF = CF = y + 10 = 20 + 10 = 30

OA is a radius. It is also the hypotenuse of a right triangle. Since chord AB ≅ to CD, then the two chords must be the same distance from the center of the circle. That means that OE ≅ OF, which is 16, so OE = 16. We know that AE ≅ DF, so AF = 30.

Use Pythagorean Therorem:

16² + 30² = (OA)²
256 + 900 = (OA)²
1156 = (OA)²
OA = 34

Not surprising since 16-30-34 is double 8-15-17, if you know your Triples.

37. If triangle RST ∼ triangle ABC, m∠A = x² - 8x, m∠C = 4x - 5, and m∠R = 5x + 30, find m∠C.

[Only an algebraic solution can receive full credit.]

Answer:

If the two triangles are similar than the corresponding angles are the congruent. That means that m∠A = m∠R.

Write a quadratic equation and solve it.

x² - 8x = 5x + 30
x² - 13x - 10 = 0
(x - 15)(x + 2) = 0
x - 15 = 0 or x + 2 = 0
x = 15 or x = -2

m∠C = 4x - 5
m∠C = 4(15) - 5 = 60 - 5 = 55, or
m∠C = 4x - 5 = 4(-2) - 5 = -13, discard this answer.

End of Part III.

More to come. Comments and questions welcome.

More Regents problems.

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