The Pythagorean Theorem

The Pythagorean theorem is the single most-used geometry fact on both the SAT and the ACT. It says that in any right triangle the two shorter sides, the legs, and the longest side, the hypotenuse, are locked together by one equation: a² + b² = c². Give a test writer any two of the three sides and the third is determined, which is exactly why so many questions hand you two lengths and ask for a third.

What makes the theorem worth a full lesson is not the formula, which you already know, but where it hides. The SAT rarely says the words right triangle. Instead it asks for the diagonal of a rectangle, the distance between two points on a coordinate plane, how far up a wall a ladder reaches, or the length of a chord in a circle, and expects you to see the right triangle inside. The ACT does the same and adds the reverse question: here are three side lengths, is the triangle right?

The two skills that separate a fast, correct answer from a slow, wrong one are naming the hypotenuse correctly before you write anything, and recognizing the handful of whole-number triples that show up again and again. This lesson builds both, then walks through the places the tests bury the triangle.

A right triangle is a triangle with one 90-degree angle. The two sides that meet at that right angle are the legs, usually written a and b, and the side opposite the right angle is the hypotenuse, written c. The hypotenuse is always the longest side, because the largest angle in any triangle sits across from the longest side and 90 degrees is the largest angle a right triangle can have. The Pythagorean theorem states that a² + b² = c²: the squares of the two legs add up to the square of the hypotenuse. The equation holds for every right triangle and for no other triangle, which is why identifying the right angle is step one of every problem.

The theorem is a statement about areas. If you build a square on each side of a right triangle, the square on the hypotenuse has exactly the same area as the two squares on the legs put together. For a 3-4-5 triangle the squares have areas 9, 16, and 25, and 9 + 16 = 25. This is also why the equation squares the sides rather than adding them directly: the lengths themselves do not add up, since 3 + 4 is not 5, but the areas of the squares do.

To find a missing hypotenuse, square both legs, add, and take the square root: $c = \sqrt{a^2 + b^2}$. To find a missing leg, square the hypotenuse, subtract the square of the known leg, and take the square root: $a = \sqrt{c^2 - b^2}$. The difference between the two is the sign in the middle, and getting it wrong is the most common error on this topic. You add when the unknown is the hypotenuse and subtract when the unknown is a leg, because the hypotenuse squared is the biggest of the three squares and the other two must be taken away from it. A side length is always positive, so you keep only the positive square root.

A Pythagorean triple is a set of three whole numbers that satisfies the equation, and the tests lean on a small family of them: 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Multiplying every side of a triple by the same number gives another triple, since scaling all three sides by k scales all three squares by k² and the equation still balances. So 6-8-10, 9-12-15, and 12-16-20 are all the 3-4-5 triangle in disguise, and 10-24-26 is 5-12-13 doubled. Recognizing a triple turns a squaring-and-rooting problem into a glance.

The distance formula is the Pythagorean theorem wearing coordinates. For two points (x₁, y₁) and (x₂, y₂), the horizontal change x₂ - x₁ and the vertical change y₂ - y₁ are the two legs of a right triangle, and the distance between the points is the hypotenuse: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Because each change is squared, it does not matter which point you call the first one or whether a change comes out negative. What does matter is subtracting correctly when a coordinate is negative, since 2 - (-4) is 6, not -2.

The theorem also runs in reverse. If the three sides of a triangle satisfy a² + b² = c² with c the longest side, the triangle must be a right triangle; if a² + b² is larger than c² the triangle is acute, and if it is smaller the triangle is obtuse. The ACT likes to ask which set of side lengths could form a right triangle, and the test is always the same: square the two smaller numbers, add, and compare with the square of the largest.

One piece of test logistics is worth knowing. The digital SAT gives you a reference sheet on every math question that lists a² + b² = c² along with the special right triangles, so you never have to trust memory for the formula itself. The ACT provides no formula sheet at all, so there the formula, the common triples, and the add-versus-subtract rule all have to come from you.

Using the wrong side as the hypotenuse. The hypotenuse is the side across from the right angle and it is always the largest number, no matter how the triangle is rotated on the page. When a triangle is drawn with its longest side flat along the bottom, students routinely call the slanted side c and get a radical instead of a whole number. Find the right angle first and read across from it.

Adding when you should subtract. When the hypotenuse is given and a leg is missing, the leg squared is c² minus the other leg squared. Adding produces a leg longer than the hypotenuse, which cannot happen. If your leg comes out bigger than c, you added.

Forgetting the square root, or forgetting to square. Legs of 6 and 8 give c² = 100, and the hypotenuse is 10, not 100. Going the other way, 6 + 8 = 14 is not the hypotenuse either, because the theorem adds the squares of the legs, not the legs. Answer choices on both tests routinely include the sum of the legs and the unrooted c² as distractors.

Splitting the radical. √(a² + b²) is not a + b, and √(16 + 36) is not 4 + 6. The square root of a sum cannot be taken term by term. Add first, then take the root of the total, and simplify by pulling out perfect-square factors: √52 = 2 √13.

Mishandling negative coordinates in the distance formula. The change in x between (-4, 3) and (2, -5) is 2 - (-4) = 6, not 2 - 4 = -2. Write the subtraction out with the parentheses in place. The sign of a change never matters because it gets squared, but the size of the change does, and a dropped negative changes the size.

Applying the theorem to a triangle that is not right. The equation a² + b² = c² holds only when there is a 90-degree angle, and the tests will happily show you a triangle with a 60-degree angle and two sides and wait for you to reach for the theorem anyway. If no right angle is given or provable, the theorem does not apply, and the question is usually about special right triangles, trigonometry, or the reverse test for whether a triangle is right at all.