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Apply Pythagorean Theorem to Find Missing Side Lengths

Students use a² + b² = c² to find the unknown side of a right triangle given the other two, in both exact form (as a square root) and approximate decimal form. They identify which side is the hypotenuse and set up the equation correctly before solving.

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Apply Pythagorean Theorem to Find Missing Side Lengths

Grade 8 · Geometry: Pythagorean Theorem · question 1 of 3

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20 rungs. A star at the top.

4 in?3 inABC

A screen's width, height and diagonal make this right triangle. How long is the screen's diagonal, in inches?

Find the missing side.

How it is taught

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In a right triangle, the legs' squares add to the hypotenuse's square.

A leg is a side that meets at the right angle.

Read it: A missing side of a right triangle

A right triangle has legs 14 and 48. The side across from the square corner is the longest. Its square is 14 × 14 + 48 × 48 = 196 + 2304 = 2500, and 50 × 50 = 2500, so the longest side is 50.

  1. Find the longest side: it is across from the square corner.
  2. For the longest side, add the squares of the legs. For a leg, take the other leg's square away from the longest side's square.
  3. Find the number that times itself makes that square.

The two leg squares together make the square on the longest side: 196 + 2304 = 2500.

So a leg's square is what is left when the other leg's square is taken away: 2500 − 196 = 2304, and 48 × 48 = 2304.

A side times itself makes its square, so the side is the number whose square it is: 50 × 50 = 2500.

The longest side: add the leg squares. A leg: take the other leg's square from the longest side's. Then find the number whose square it is.

Adding the legs: 14 + 48 = 62.

Add their squares, not the legs: 196 + 2304 = 2500, and 50 × 50 = 2500, so the longest side is 50.

A right triangle has legs 5 and 12. Its longest side is ___. — 13. 5 × 5 = 25 and 12 × 12 = 144: 25 + 144 = 169, and 13 × 13 = 169.

Find c when legs are 5 and 3

Here's a right triangle. We know two sides — five and three — and we want to find the long one.

Grade 8 · Right triangle with legs 5 and 3

The longest side, across from the right angle, is called the hypotenuse. Let's call it c.

The Pythagorean theorem says: a squared plus b squared equals c squared.

Our legs are five and three. Plug them in: five squared plus three squared equals c squared.

Five squared is twenty-five. Three squared is nine.

Twenty-five plus nine is thirty-four. So c squared equals thirty-four.

To find c, take the square root of both sides. c equals the square root of thirty-four — that's our exact answer.

On a calculator, the square root of thirty-four is about five point eight three.

So c equals root thirty-four, about five point eight three units. Nice work!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
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Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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