Question review: the circle through three points, and the one time the hand method wins

Third in the series. The first two were questions where Desmos won clearly. This one is more interesting, because it is a question where the calculator is fast, the general hand method is slow, and there is a third way that beats both, if you see it. Which method is best depends on how fast you can tell which question you are looking at.

The question

In the xy-plane, a circle passes through the points (0, 0), (6, 0) and (0, 8). What is the radius of the circle?

A box to type into. Geometry and Trigonometry, usually in the second half of a module. The cousins ask for the center, or for the equation, or give the three points in a word problem about a round garden or a sprinkler.

By hand, the general way

Three points determine a circle, and the center is where the perpendicular bisectors of any two chords meet.

The chord from (0, 0) to (6, 0) is horizontal with midpoint (3, 0), so its perpendicular bisector is the vertical line x = 3. The chord from (0, 0) to (0, 8) is vertical with midpoint (0, 4), so its bisector is y = 4. The center is (3, 4). The radius is the distance from the center to any of the points: from (3, 4) to (0, 0) is √(9 + 16) = 5.

That took about sixty seconds and was easy only because two of the chords were on the axes. With three arbitrary points the bisectors have slopes and intercepts, and the same method takes two or three minutes and a page of arithmetic, which on this test means it is not an option.

With Desmos

Click the plus button, choose Table, and enter the three points in x₁ and y₁. On the next line type the circle’s equation with a tilde instead of an equals sign:

(x_1-h)^2+(y_1-k)^2 ~ r^2

Desmos fits it and lists the parameters: h = 3, k = 4, r = 5. Below them it shows RMSE, which is zero, which is the calculator telling you the fit is exact. If RMSE is not zero on a three-point circle, a coordinate was typed wrong. Radius 5.

Time: about twenty seconds, if the tilde regression is already a habit. It works for any three points, arbitrary or not, and it never gets harder. This is the method to know for the word-problem versions of this question, and it is also the method for the parabola through three points, with a different line of code.

Two traps. Desmos may report r as negative, because r² is what was fit; take the absolute value. And it will not fit h, k or r if any of those letters is already a slider from the previous question, so clear the list first.

By hand, the short way

Look at the three points again. (0, 0), (6, 0) and (0, 8) make a right triangle with the right angle at the origin, legs 6 and 8. A right angle inscribed in a circle sits on a diameter, so the hypotenuse is a diameter. The hypotenuse is 10, the radius is 5.

Time: ten seconds, no calculator, no algebra. It also gives the center for free, as the midpoint of the hypotenuse, (3, 4).

Which is faster

The short hand method, by a lot, but only on this version of the question. It depends on noticing a right angle, and the SAT writes this question with a right angle in it more often than chance would suggest, because the geometry is what it is testing. With three points that do not make a right angle, the shortcut is gone and Desmos wins by two minutes.

So the real skill is the five-second decision at the start. Look at the points. Is there an obvious right angle, or a chord that is clearly a diameter? If yes, the hand method. If no, or if you are not sure within five seconds, the table and the tilde. The worst outcome is the one I see most in logs: the student starts the general hand method, gets a minute in, realizes it is going to be long, and then opens Desmos, having paid for both.

What it is actually testing

Two things. Whether you know that three points fix a circle and roughly why. And whether you know the inscribed-angle fact, that a right angle on a circle subtends a diameter, which is on the test in several costumes and in the reference sheet in none of them. The regression gets you the point without either. The shortcut gets you the point with both, faster.

Learn the regression first, because it always works. Then learn to look for the triangle, because when it is there, it is the fastest thing on the section.