Pythagorean Theorem on the SAT
The Right Triangle Shortcut
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Pythagorean Theorem
The Pythagorean Theorem is the fastest tool for right triangles. Any time you see a right angle, you should think $a^2 + b^2 = c^2$. The SAT expects you to use it quickly and accurately.
This lesson shows how to identify the hypotenuse, plug in the legs correctly, and connect the theorem to distance on the coordinate plane. You will also see how to avoid the common error of squaring the hypotenuse incorrectly.
A Simple Definition Unlocks Pythagorean Theorem
The hypotenuse is always the side opposite the right angle, and it is always the longest side. The theorem states that the sum of the squares of the legs equals the square of the hypotenuse.
In coordinate geometry, the distance formula is just the Pythagorean Theorem in disguise. If you can compute the horizontal and vertical change, you can find the distance between points without memorizing a new formula.
Work Through Pythagorean Theorem Step by Step
A right triangle has legs $6$ and $8$. What is the hypotenuse?
Find the hypotenuse when the legs are $6$ and $8$, then check that it is the longest side.
Apply the Pythagorean Theorem to connect the legs and hypotenuse.
Compute the result to simplify the expression.
Simplify the expression to make the next step clear.
Take the square root to undo the square.
Use Desmos to Check Pythagorean Theorem
A right triangle has legs $6$ and $8$. What is the hypotenuse?
Desmos can solve Pythagorean problems quickly by calculating squares and square roots. It is useful when the numbers are not perfect squares.
sqrt(6^2 + 8^2)
Algebra is fast for clean triples. Desmos is faster for messy decimals.
Expert move: Use Desmos to square and take roots when the arithmetic is messy, but set up $a^2 + b^2 = c^2$ first.
When to skip Desmos: For common triples, mental math is faster.
- Desmos features used: arithmetic evaluation.
- Common mistake: squaring the hypotenuse instead of taking the square root at the end.
Practice Pythagorean Theorem with SAT-Style Questions
Apply $a^2 + b^2 = c^2$.
A right triangle has legs and . What is the hypotenuse?
A right triangle has hypotenuse and one leg . What is the other leg?
A ladder is ft long and reaches a window ft from the ground. How far is the base from the wall?
If the legs of a right triangle are and , what is the area?
Key Takeaways to Remember for Pythagorean Theorem
- Identify the hypotenuse as the side opposite the right angle.
- Use $a^2 + b^2 = c^2$ and solve for the missing side.
- Desmos helps when square roots are not perfect squares.
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