Pythagoras' Theorem
Pythagoras' theorem says that in a right-angled triangle, a² + b² = c². The two shorter sides squared add up to the longest side squared, so 3² + 4² = 5². That longest side, c, is the hypotenuse, opposite the right angle.
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Pythagoras' Theorem
- Only works in a right-angled triangle
- The square of the hypotenuse equals the sum of the squares of the other two sides:
Finding a Missing Side in a Triangle
- Identify the hypotenuse first (the side opposite the right angle)
- Substitute the known lengths into , then solve
Finding the Distance Between Two Points
- Draw a right-angled triangle using horizontal and vertical distances
- Use Pythagoras to find the diagonal distance:
Using Pythagoras in 3D Shapes
- Break the problem into two right-angled triangles
- Apply Pythagoras twice to find the longest diagonal
Practice Questions
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What is the length of the hypotenuse in a right-angled triangle with sides 3 and 4?
Correct! 🎉 +10 pointsNot quite right
Applying Pythagoras' theorem: , where and . We set the hypotenuse as x and solve , which gives , so . Taking the square root of both sides, we get .
What is the length of the hypotenuse in a right-angled triangle with sides 6 and 8?
Correct! 🎉 +10 pointsNot quite right
We use Pythagoras' theorem: , where and . We set the hypotenuse as x and solve , which gives , so . Taking the square root of both sides, we get .
What is the length of the hypotenuse in a right-angled triangle with sides 9 and 12?
Correct! 🎉 +20 pointsNot quite right
Applying Pythagoras' theorem: , where and . We set the hypotenuse as x and solve , which gives , so . Taking the square root of both sides, we get .
In a right-angled triangle, the sides are 12, 16, and 20. Which is the hypotenuse?
Correct! 🎉 +20 pointsNot quite right
The hypotenuse is always the longest side in a right-angled triangle. Here, the longest side is 20, so it must be the hypotenuse.
In a right-angled triangle, the hypotenuse is 13, and one of the shorter sides is 12. What is the length of the other shorter side?
Correct! 🎉 +20 pointsNot quite right
Using Pythagoras' theorem: , where and . We set the missing side as x and solve , which gives , so . Taking the square root of both sides, we get .
What is the distance between the points (2, 2) and (8, 10) on a coordinate plane?
Correct! 🎉 +30 pointsNot quite right
We form a right-angled triangle using the horizontal and vertical distances. The horizontal distance is , and the vertical distance is . We set the unknown distance as x and apply Pythagoras' theorem: , which simplifies to . We get , and .
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Students Also Ask
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Pythagoras' theorem is written as a² + b² = c². Here c is the hypotenuse, and a and b are the two shorter sides. It tells you that the square of the hypotenuse equals the sum of the squares of the other two sides.
Yes. In a right-angled triangle, the hypotenuse is always the longest side, and it always sits opposite the right angle. Because it is the longest side, it takes the place of c in the formula a² + b² = c².
You can use Pythagoras' theorem only in a right-angled triangle. That is a triangle containing a right angle of 90°. If the triangle has no right angle, the theorem does not apply, so always check for the right angle first.
When you take the square root at the end, you get two solutions, one positive and one negative. A length or distance cannot be negative, so you reject the negative solution and keep the positive value as your final answer.
Form a right-angled triangle using the horizontal and vertical gaps between the two points as the shorter sides. The distance between the points is the hypotenuse. Apply Pythagoras' theorem to the two gaps, then take the positive square root to find the distance.
Yes. To find the space diagonal of a cuboid, you use two right-angled triangles. First find the diagonal across the base. Then use that diagonal with the height as the two sides of a second triangle. Pythagoras' theorem then gives the space diagonal.