Geometry PARCC question: diagonal length

The following multiple-select question, explained here in hopes of helping geometry students in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2015 test in geometry, here:

A square has sides that are each 90 feet long. Which equations can be used to calculate d, the length of a diagonal of the square, in feet?

Select all that apply.

  1.     d = \frac{1}{2}(90 \times 90)
  2.     d = \sqrt{90^2+90^2}
  3.     d = \sqrt{4 \times 90^2}
  4.     \cos 45^{\circ} = \frac{90}{d}
  5.     \sin 45^{\circ} = \frac{90}{d}
  6.     \tan 45^{\circ} = \frac{90}{d}
Answer and references

Correct answers: 2, 4, and 5 all can be used to calculate the diagonal.

PARCC evidence statement(s) tested: G-SRT.8:

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

The task may have a real world or mathematical context. For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required; however, students will not be penalized if they simplify the radicals correctly.

The evidence statement above references Math Practice 1, Math Practice 2, Math Practice 5, and Math Practice 6 in the Common Core. (Follow the links for the complete text of the standards of mathematical practice.)

The question tests students’ understanding of the high school Common Core geometry standard HSG.SRT.C.8, found under High School: Geometry, Similarity, Right Triangles, & Trigonometry (define trigonometric ratios and solve problems involving right triangles), which states that they should be able to “understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.” The first three options test the eighth-grade geometry standard 8.G.B.7 (understand and apply the Pythagorean Theorem), which says students should be able to “apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.”

Explanation of each answer choice
square with side 90 feet and unknown diagonal

Option 1 is incorrect. That’s how you would find the area for half the square, either the upper or lower half, but not the length of the diagonal.

Option 2 is correct. Because a2 + b2 = c2, and the diagonal forms the hypotenuse of two isosceles right triangles with legs of length 90 feet, we can solve for c:

a^2 + b^2 = c^2
\sqrt{a^2 + b^2} = c

Option 3 is incorrect. I suppose 4 is 22, but I don’t know where that comes from either. Four sides in a square, maybe?? In any event, it’s wrong.

Option 4 is correct. The cosine of an angle in a right triangle is the ratio of the side adjacent to the angle and the hypotenuse. The hypotenuse is d, and the diagonal splits the 90° angles in the square into two 45° angles.

Option 5 is correct. The sine of an angle in a right triangle is the ratio of the side opposite that angle and the hypotenuse. The same reasoning applies here as in option 4.

Option 6 is incorrect. The tangent would be the opposite side divided by the adjacent side. The diagonal, or hypotenuse, is in the denominator, which is where you would put the adjacent side if you were using the tangent. The hypotenuse is just the hypotenuse. It is never the adjacent side; it is never the opposite side; it’s just the hypotenuse.

Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a few pages about the Pythagorean Theorem in case you need to review your eighth-grade geometry, beginning here.

The Khan Academy, developed by Sal Khan, an engineer who has a few videos dealing with how to use trigonometry to find the sides of a right triangle. With trigonometry, you can even find the length of two sides given one side and one of the acute angles in a right triangle.

Chapter 9 of the book Geometry for Enjoyment and Challenge by Richard Rhoad et al, all teachers from Illinois, puts it all together, including solving the Pythagorean Theorem equation for any side in a right triangle given the other two, and using trig to find the length of any side in a right triangle given one side and one of the acute angles.

(Richard Rhoad, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge, new edition. Evanston, Ill.: McDougal Littell, a division of Houghton Mifflin Company, 1991. The book is used in several geometry classes taught in Illinois high schools.)

Analysis of this question and online accessibility

The question measures knowledge of the Common Core standards it purports to measure, one being in eighth grade and the other in high school. It tests students’ ability apply the Pythagorean Theorem to find the distance of the hypotenuse of a right triangle and to use trigonometric ratios to find it. It is considered to have a low cognitive demand.

The question can be tested online and should yield results that are as valid and reliable as those obtained on paper.

No special accommodation challenges can be identified with this question, so the question is considered fair.

Challenge

Often these problems will have a real-world context, rather than just being about a square. Pretend you are the groundskeeper at a stadium and you have to order a new rope for one of the flagpoles. To find out what length of rope you need, you notice that the flagpole casts a shadow 11.6 meters long on the ground. The angle of elevation of the sun is 36°50′ and you need to know how tall the pole is. Then, you’ll need to double that to order the rope and add a little slack. What length of rope will you buy?

Purpose of this series of posts

Voxitatis is developing blog posts that address every algebra 1 question released to the public by the Partnership for Assessment of Readiness for College and Careers, or PARCC, in order to help students prepare to take the test this spring.

Our total release will run from February 27 through March 15, with one or two questions discussed per day. Then we’ll move to geometry at the end of March, algebra 2 during the first half of April, and eighth grade during the last half of April.

About the Author

Paul Katula
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.