ACARA v9 CONTENT DESCRIPTION “use the Pythagorean theorem to find the unknown side lengths of right-angled triangles, and to test whether a triangle is right-angled”
The sides of a right-angled triangle
A right-angled triangle is a triangle with one right angle, the square corner you see where two walls meet. The two sides that form that corner are called the legs, and we label them a and b. The third side, the one that sits directly opposite the right angle, is the hypotenuse, written c. The hypotenuse is always the longest side of a right-angled triangle, because the largest angle in any triangle faces the longest side, and the right angle is the largest angle here. Before doing any calculation it helps a student to name the three sides correctly, since every later step depends on knowing which side is the hypotenuse and which two are the legs.
The parts of a right-angled triangle
Legs a and b, hypotenuse c opposite the right angle.
in a right-angled triangle the two shorter sides are the legs a and b, and the hypotenuse c is opposite the right angle and is the longest side.
The Pythagorean theorem
For every right-angled triangle the three sides obey a single relationship: a^2 + b^2 = c^2. In words, the square of the hypotenuse equals the sum of the squares of the two legs. A picture makes this believable. If you draw an actual square on each side of the triangle, the area of the square on the hypotenuse is exactly equal to the combined area of the squares on the two legs. With legs of 3 and 4 the leg squares have areas 9 and 16, the hypotenuse square has area 25, and 9 + 16 = 25. This is the theorem of Pythagoras, and it holds only when the triangle has a right angle.
Squares on the three sides
The square on the hypotenuse equals the two leg squares.
the square on the hypotenuse equals the sum of the squares on the legs: a^2 + b^2 = c^2, here 9 + 16 = 25.
Finding the hypotenuse
When both legs are known and the hypotenuse is the unknown, no rearranging is needed: simply add the squares of the legs and take the square root, so c = sqrt(a^2 + b^2). Take a triangle with legs 6 and 8. Squaring gives 36 and 64, and adding gives 100, so c = sqrt(100) = 10. The hypotenuse is the side opposite the right angle, and it must come out longer than either leg, which is a quick way to check that an answer is sensible. Here 10 is indeed longer than both 6 and 8, so the result is reasonable.
Finding the hypotenuse
Add the leg squares, then take the square root.
to find the hypotenuse, add the squares of the legs and take the square root: sqrt(6^2 + 8^2) = sqrt(100) = 10.
Finding a shorter side
Sometimes the hypotenuse is known and one of the legs is missing. Now the same relationship is rearranged: starting from a^2 + b^2 = c^2 and subtracting a^2 from both sides gives b^2 = c^2 - a^2, so b = sqrt(c^2 - a^2). With a hypotenuse of 13 and one leg of 5, square to get 169 and 25, subtract to get 144, and take the root to find b = sqrt(144) = 12. The key habit is to subtract, not add, whenever the hypotenuse is the known side, because a leg must always be shorter than the hypotenuse.
Finding a shorter side
Subtract from the hypotenuse square, then take the root.
to find a leg, subtract the square of the known leg from the square of the hypotenuse, then take the root: sqrt(13^2 - 5^2) = sqrt(144) = 12.
Testing for a right angle
The relationship also works in reverse, and this reverse is called the converse. A triangle is right-angled exactly when a^2 + b^2 = c^2, with c the longest side. To test a set of three lengths, square them, add the two smaller squares, and see whether the total matches the square of the longest side. For 6, 8 and 10, 36 + 64 = 100 and 10^2 = 100, so the triangle is right-angled. For 6, 8 and 11, 36 + 64 = 100 but 11^2 = 121, and 100 is not 121, so that triangle is not right-angled. Whole-number sets that pass the test, such as 3, 4, 5 and 5, 12, 13 and 6, 8, 10, are known as Pythagorean triples.
Testing for a right angle
Use the converse: does a^2 + b^2 = c^2?
a triangle is right-angled only if a^2 + b^2 = c^2; 6, 8, 10 passes (100 = 100) but 6, 8, 11 fails (100 is not 121).
Why this matters
The theorem of Pythagoras ties together the three sides of every right-angled triangle, and it is one of the most used results in all of mathematics. Builders rely on it to check that a corner is truly square, navigators and designers use it to find straight-line distances, and the sizes of screens, ramps and roof braces are all worked out with it. The two slips to watch for are forgetting that the hypotenuse is the longest side, and adding the squares when the problem actually calls for subtracting one. This unit stays with right-angled triangles and whole-number arithmetic; the related angle work of trigonometry, with sine, cosine and tangent, comes in a later year.
Quick self-check
1. In a right-angled triangle, which side is the hypotenuse?
2. What is the Pythagorean theorem for a right-angled triangle with legs a, b and hypotenuse c?
3. A right-angled triangle has legs 6 and 8. How long is the hypotenuse?
4. A right-angled triangle has hypotenuse 13 and one leg 5. How long is the other leg?
5. Is a triangle with sides 6, 8 and 11 right-angled?