In the quarter circle above, y in terms of x

2018 Handbook

2018 Test B Math · Question 43 of 57

Also in 2026 Practice Test B Math, question 36; 2025 Practice Test B Math, question 43.

Question

$A quarter circle whose center, marked with a dot, is at the bottom left corner. A segment labeled x + 1 runs from the center to a point on the arc, and from that point a vertical segment drops to the bottom edge, meeting it at a right angle. Along the bottom edge, the part from the center to the foot of that segment is labeled y, and the part from the foot to the end of the arc is labeled 2.$

In the quarter circle above, what is $y$ in terms of $x$ ?

A $x-1$
B $x+1$
C $\frac{x+1}{2}$
D $\sqrt{\frac{\left ( x+1 \right )^{2}}{2}}$

Show the answer

Correct answer: A

Both $x+1$ and $y+2$ are radii.

So, set them equal to each other and solve for $y$ .

$y+2=x+1$

$y=x-1$

Question analysis

In a quarter circle, one segment is labeled x + 1 and the bottom edge is split into lengths y and 2, and students must write y in terms of x. It tests basic circle facts together with solving a simple equation. The challenge is seeing which marked lengths must be equal.

NYS standards: 7.G.4 circle area and circumference
7.EE.4 equations and inequalities from problems
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Grade 7 circle work centers on area and circumference formulas. This needs only the definition of a radius and a one-step equation, but spotting the relationship in a figure is less routine than plugging into a formula.
Estimated difficulty: Moderate: several steps, or a close reading with one strong distractor.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly analyze
What makes it hard: Reading a figure, table or graph major
Variables instead of numbers minor
A very close wrong answer minor
Solution: 2 steps
Reading level: Question and choices: about grade 9, 16 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The figure looks like a right-triangle problem, and that is the trap: the right angle and the slanted radius pull students toward the Pythagorean theorem, which cannot work without the vertical leg. The solution needs only the fact that all radii of a circle are equal, applied to the bottom edge. Students who mark every segment that starts at the center see it at once.

Reading a figure, table or graph major Understanding the question
The labels sit on two different radii: x + 1 on the slanted one, and y + 2 along the bottom edge, which runs from the center all the way to the arc. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Variables instead of numbers minor Understanding the question
The answer is an expression in x, so there is no number to check; trying x = 5, which makes the radius 6 and y = 4, is the fallback. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)
A very close wrong answer minor Choosing the answer
x + 1 differs from the right x − 1 by a single sign; only seeing that y plus the 2-unit piece makes a radius, so the 2 is subtracted, separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Look for two lengths in the figure that are both radii of the quarter circle, set their expressions equal, and solve for y.

Traps

Trusting the picture: The figure suggests something it does not state; work from the given facts.
Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.

The wrong answers, classified

B Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals partial knowledge Expected pull: moderate
Sets y equal to the whole radius x + 1, ignoring the 2-unit piece that completes the bottom radius beyond y.
C Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)
Reveals a surface strategy Expected pull: weak
Treats y and the 2 as factors, y × 2 = x + 1, and divides by 2 where it should subtract 2.
D Trusting the picture (Lesh, Post & Behr, 1987)
Reveals a misconception Expected pull: strong
Trusts the drawing's nearly 45° radius, assumes the triangle's two legs are equal, and solves y² + y² = (x + 1)² for y.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Radii of a circle are equal: $r_1 = r_2$; A radius is a segment from the center of a circle to its curved edge. r₁ and r₂ stand for any two radii of the same circle. They are always equal, because every point on the curve is the same distance from the center.

Read the figure
The shape is a quarter circle, one fourth of a whole circle. Its center is the corner with the dot. A radius is a segment from the center to the curved edge. The slanted segment, labeled $x+1$ , goes from the center to the curve. The bottom edge is split into two parts, $y$ and 2.
Spot the two radii
The slanted segment is a radius, so the radius is $x+1$ . The whole bottom edge also runs from the center to the curve. Its length is $y+2$ , so it is a radius too.
Watch out: the slanted radius only looks like it is at 45 degrees. Do not assume the triangle’s two short sides are equal. That idea leads to choice D.
Set the radii equal
Use the rule that all radii of a circle are equal.
$y + 2 = x + 1$
Now you have one equation with $x$ and $y$ .
Solve for y
Get $y$ alone by taking 2 from both sides.
$y = x + 1 - 2 = x - 1$
Watch out: $y$ is only part of the bottom radius. Setting $y$ equal to $x+1$ gives choice B. Dividing by 2 instead of subtracting 2 gives choice C.
Give the answer
$y$ is 2 less than the radius, and the radius is $x+1$ . So the answer is A, $x-1$ .

Check your answer

Pick a number. If $x = 5$ , the radius is 6. Then $y = 5 - 1 = 4$ , and $4 + 2 = 6$ , which matches the radius. Also, $y$ is a short side of the right triangle. It should be shorter than the slanted side, and 4 is less than 6.

The original DOE question

Question 100 as printed on page 161 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 161, question 100. Open page 161 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).