Pythagoras - Corbettmaths

Pythagoras Theorem

In this section, the instructor explains Pythagoras's theorem and how it is used to find the length of a missing side in a right-angled triangle.

Understanding Pythagoras's Theorem

• Pythagoras's theorem is used to work out the length of a missing side in a right-angled triangle.

• The theorem states that a squared plus b squared equals c squared, where c is the hypotenuse (longest side), and a and b are the two shorter sides.

• To use Pythagoras's theorem, you need to square the two short sides, add them together, and then take the square root of that sum to get the length of the longest side.

Applying Pythagoras's Theorem

• To find the longest side (hypotenuse), you need to label each side as A, B or C. Labeling C as the longest side is important.

• Example 1: If one short side is 3 cm and another short side is 4 cm, then using Pythagoras's theorem we can calculate that the hypotenuse (longest side) is 5 cm.

• Example 2: If one short side is 6 cm and another short side is unknown (labeled as x), then using Pythagoras's theorem we can calculate that x = 10 cm.

• Example 3: If one short side is 9 cm and hypotenuse (longest side) is known to be 16 cm, then using Pythagoras's theorem we can calculate that the other short side is 13 cm.

Solving for Short Sides of Right-Angled Triangles

In this section, the instructor explains how to use Pythagoras's theorem to solve for the short sides of right-angled triangles.

Using Pythagoras's Theorem

• To solve for the short sides of a right-angled triangle, you can use Pythagoras's theorem: a² + b² = c².

• To solve for one of the short sides (let's call it "a"), rearrange the equation to get a² = c² - b². Then take the square root of both sides to get a = √(c² - b²).

• For example, if we have a triangle with hypotenuse c = 35 and one short side b = 17, we can solve for the other short side a as follows:

• Square both c and b: c² = 1225 and b² = 289.

• Substitute into Pythagoras's theorem: a² + 289 = 1225.

• Rearrange to isolate a: a² = 936.

• Take the square root of both sides: a ≈ 30.59 cm.

Finding Shorter Sides Using Practice Problems

In this section, the instructor explains an alternative method for finding shorter sides using practice problems.

Squaring and Subtracting Method

• Another way to find shorter sides is by squaring both longer and shorter sides, subtracting them from each other, and then taking the square root of that difference.

• For example, if we have a triangle with longer side A = 81 cm and shorter side B = x, we can solve for x as follows:

• Square both A and B: A² = 6561 and B² = x².

• Subtract B² from A²: 6561 - x².

• Set the difference equal to the given value (e.g. 175): 6561 - x² = 175.

• Rearrange to isolate x: x² = 6386.

• Take the square root of both sides: x ≈ 79.93 cm.

Conclusion

In this section, the instructor concludes the lesson on solving for short sides of right-angled triangles.

• Remember that Pythagoras's theorem is a useful tool for solving for short sides of right-angled triangles.

• Alternatively, you can use the squaring and subtracting method to find shorter sides using practice problems.