BackClose

Select a Subject

Subject Test Math 2

Question 22 of 28

beginning of content:
Tags:
SAT Subject Test

What is the measure of one of the larger angles of a parallelogram in the x y minus text plane end text that has vertices with coordinates left parenthesis 2 comma 1 right parenthesis, left parenthesis 5 comma 1 right parenthesis, left parenthesis 3 comma 5 right parenthesis and left parenthesis 6 comma 5 right parenthesis?

Select an Answer

93.4 straight degree

Correct Answer: 
No

96.8 straight degree

Correct Answer: 
No

104.0 straight degree

Correct Answer: 
Yes

108.3 straight degree

Correct Answer: 
No

119.0 straight degree

Correct Answer: 
No

The correct answer is C.

First, note that the angle of the parallelogram with vertex left parenthesis 3 comma 5 right parenthesis is one of the two larger angles of the parallelogram: Looking at the graph of the parallelogram in the x y minus text plane end text makes this apparent. Alternatively, the sides of the angle of the parallelogram with vertex left parenthesis 3 comma 5 right parenthesis are a horizontal line segment with endpoints left parenthesis 3 comma 5 right parenthesis and left parenthesis 6 comma 5 right parenthesis comma and a line segment of positive slope with endpoints left parenthesis 2 comma 1 right parenthesis and left parenthesis 3 comma 5 right parenthesis that intersects the horizontal line segment at its left endpoint left parenthesis 3 comma 5 right parenthesis, so the angle must measure more than 90 straight degree. Since the sum of the measures of the four angles of a parallelogram equals 360 straight degree, the angle with vertex left parenthesis 3 comma 5 right parenthesis must be one of the larger angles.

One way to determine the measure of the angle of the parallelogram with vertex left parenthesis 3 comma 5 right parenthesis is to apply the law of cosines to the triangle with vertices left parenthesis 2 comma 1 right parenthesis, left parenthesis 3 comma 5 right parenthesis, and left parenthesis 6 comma 5 right parenthesis. The length of the two sides of the angle with vertex left parenthesis 3 comma 5 right parenthesis are square root of left parenthesis 3 minus 2 right parenthesis to the power of 2 end exponent plus left parenthesis 5 minus 1 right parenthesis to the power of 2 end exponent end root equals square root of 17 end root and square root of left parenthesis 3 minus 6 right parenthesis to the power of 2 end exponent plus left parenthesis 5 minus 5 right parenthesis to the power of 2 end exponent end root equals 3; the length of the side opposite the angle is square root of left parenthesis 6 minus 2 right parenthesis to the power of 2 end exponent plus left parenthesis 5 minus 1 right parenthesis to the power of 2 end exponent end root equals 4 square root of 2 end root. Let theta represent the angle with vertex left parenthesis 3 comma 5 right parenthesis and apply the Law of Cosines: A to the power of 2 end exponent plus B to the power of 2 end exponent minus 2 A B cos theta equals C to the power of 2 end exponent, so cos theta equals fraction numerator C to the power of 2 end exponent minus left parenthesis A to the power of 2 end exponent plus B to the power of 2 end exponent right parenthesis over denominator negative 2 A B end fraction equals fraction numerator 32 minus left parenthesis 17 plus 9 right parenthesis over denominator negative 2 left parenthesis square root of 17 end root right parenthesis left parenthesis 3 right parenthesis end fraction equals fraction numerator 6 over denominator negative 6 square root of 17 end root end fraction equals negative fraction numerator 1 over denominator square root of 17 end root end fraction. Therefore, the measure of one of the larger angles of the parallelogram is arccos function application left parenthesis negative fraction numerator 1 over denominator square root of 17 end fraction right parenthesis almost equal to 104.0 straight degree.

Another way to determine the measure of the angle of the parallelogram with vertex left parenthesis 3 comma 5 right parenthesis is to consider the triangle left parenthesis 2 comma 1 right parenthesis, left parenthesis 3 comma 5 right parenthesis, and left parenthesis 3 comma 1 right parenthesis. The measure of the angle of this triangle with vertex left parenthesis 3 comma 5 right parenthesis is 90 straight degree less than the measure of the angle of the parallelogram with vertex left parenthesis 3 comma 5 right parenthesis. The angle of the triangle has opposite side of length 3 minus 2 equals 1 and adjacent side of length 5 minus 1 equals 4, so the measure of this angle is arctan function application fraction numerator 1 over denominator 4 end fraction. Therefore, the measure of the angle of the parallelogram with vertex left parenthesis 3 comma 5 right parenthesis is 90 straight degree plus arctan function application 1 fourth almost equal to 104.0 straight degree.

Yet another way to determine the measure of the angle of the parallelogram with vertex left parenthesis 3 comma 5 right parenthesis is to use trigonometric relationships to find the measure of one of the smaller angles, and then use the fact that each pair of a larger and smaller angle is a pair of supplementary angles. Consider the angle of the parallelogram with vertex left parenthesis 2 comma 1 right parenthesis; this angle coincides with the angle at vertex left parenthesis 2 comma 1 right parenthesis of the right triangle with vertices at left parenthesis 2 comma 1 right parenthesis, left parenthesis 3 comma 5 right parenthesis, and left parenthesis 3 comma 1 right parenthesis, with opposite side of length 5 minus 1 equals 4 and adjacent side of length 3 minus 2 equals 1, so the measure of this angle is arctan invisible times 4. This angle, together with the angle of the parallelogram with vertex left parenthesis 3 comma 5 right parenthesis, form a pair of interior angles on the same side of a transversal that intersects parallel lines, so the sum of the measures of the pair of angles equals 180 straight degree. Therefore, the measure of the angle of the parallelogram with vertex left parenthesis 3 comma 5 right parenthesis is 180 straight degree minus arctan space 4 almost equal to 104.0 straight degree.

Question Difficulty: 
hard