The Pythagorean Theorem

1. The Fundamental Theorem

The Pythagorean Theorem states that in any right triangle (a triangle with one $90^\circ$ angle), the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the two legs ($a$ and $b$).

The formula is expressed as: $a^2 + b^2 = c^2$

When it appears: This is the go-to tool whenever you are given two sides of a right triangle and need the third. How to recognize it: Look for the "square" symbol in the corner of a triangle. If the SAT doesn't explicitly say it's a right triangle, look for keywords like "perpendicular," "vertical and horizontal," or "tangent to a circle."

SAT Reference Sheet Note: This formula is provided on the SAT reference sheet. However, if you have to look at the reference sheet to remember $a^2 + b^2 = c^2$, you are losing valuable time. You should have this memorized until it is second nature.

2. Identifying the Hypotenuse

The most common error students make is plugging the wrong numbers into the wrong variables.

• Legs ($a$ and $b$): The two sides that form the $90^\circ$ angle.
• Hypotenuse ($c$): The longest side, located directly across from the right angle.

On the Digital SAT, triangles are often rotated or flipped. Always locate the $90^\circ$ angle first; the side it "points to" is always $c$.

3. Pythagorean Triples (The "Time Savers")

The SAT is a timed test. While you can always use the formula, the test makers frequently use "Triples"—sets of three integers that perfectly satisfy the theorem. If you recognize these, you can solve a question in 3 seconds instead of 30.

The Big Two:

1. 3-4-5 Triangle: $3^2 + 4^2 = 5^2 \rightarrow 9 + 16 = 25$.
2. 5-12-13 Triangle: $5^2 + 12^2 = 13^2 \rightarrow 25 + 144 = 169$.

The Multiples: The SAT loves to use multiples of these triples. For example:

• 6-8-10 (a 3-4-5 multiplied by 2)
• 9-12-15 (a 3-4-5 multiplied by 3)
• 30-40-50 (a 3-4-5 multiplied by 10)
• 10-24-26 (a 5-12-13 multiplied by 2)

Other Triples to Know:

• 8-15-17
• 7-24-25

4. The Distance Formula Connection

One of the most common ways the SAT tests the Pythagorean Theorem is through the Distance Formula in the $xy$-plane. The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

If you look closely, this is just the Pythagorean Theorem rearranged: $d^2 = (\Delta x)^2 + (\Delta y)^2$ Where $\Delta x$ is the horizontal leg and $\Delta y$ is the vertical leg. On the Digital SAT, if you are asked for the distance between points, you can draw a right triangle on the screen (or in your head) and use $a^2 + b^2 = c^2$.

5. Hidden Right Triangles

The SAT rarely gives you a simple triangle and asks for $x$. They hide right triangles inside other shapes:

• Isosceles Triangles: Drawing an altitude (a line from the top vertex to the base) creates two congruent right triangles.
• Rectangles/Squares: The diagonal of a rectangle creates two right triangles.
• Circles: A radius drawn to a tangent line always forms a $90^\circ$ angle. Also, any triangle inscribed in a semicircle where one side is the diameter is a right triangle.

1. Spot the Triple First

Before you reach for your calculator, look at the sides. If you see a leg of 6 and a hypotenuse of 10, don't do $10^2 - 6^2$. Recognize it immediately as a 6-8-10 triangle. The answer is 8. This saves mental energy for the harder questions at the end of the module.

2. Use the Desmos Calculator for Radicals

On the Digital SAT, the built-in Desmos calculator is your best friend. If you calculate $c = \sqrt{149}$ and the answer choices are in decimal form, use Desmos. If the answer choices are in simplified radical form (like $5\sqrt{2}$), you can type $5\sqrt{2}$ into Desmos to see if its decimal value matches your $\sqrt{50}$.

3. Draw the "Hidden" Line

If a problem shows a trapezoid or an irregular quadrilateral, look for a way to drop a vertical line to create a rectangle and a right triangle. This is a classic SAT move to turn a "hard" geometry problem into a "foundational" Pythagorean problem.

4. Don't Over-Calculate

If the question asks for the area of a square built on the hypotenuse, and you know $a^2 + b^2 = c^2$, remember that $c^2$ is the area of that square. You don't need to find $c$ and then square it again.