Pythagoras Theorem — NSW Year 9 Advanced Maths
Right-angled triangles and surds, from exact lengths to problems in three dimensions. Nine subtopics and 180 practice questions with fully worked solutions, aligned to NESA outcomes MA4-PYT-C-01 and MA5-IND-P-02.
Pythagoras Theorem in NSW Year 9 Advanced
Pythagoras' theorem links the three sides of any right-angled triangle: \(c^2 = a^2 + b^2\), where \(c\) is the hypotenuse. Year 9 Advanced revises the theorem from Stage 4 and takes it further with surds, the exact roots such as \(5\sqrt{2}\) that many answers produce, and with problems in solids such as pyramids, cones and boxes.
The nine subtopics below find the hypotenuse and a shorter side, work through surds and irrational numbers, simplifying surds, and adding, subtracting, multiplying and dividing them, then return to problems, tests for right-angled triangles and Pythagorean triads. Pythagoras Theorem is one of the 14 topics in NSW Year 9 Advanced Maths and belongs to NESA's Measurement and Space strand. It revises NESA outcome MA4‑PYT‑C‑01, and the surd work comes from MA5‑IND‑P‑02. Right-angled triangles lead straight into trigonometry and the distance formula on the number plane.
Subtopics
Pythagoras Theorem: common questions
How is Pythagoras' theorem used in Year 9 Advanced maths?
It finds a missing side of a right-angled triangle, tests whether a triangle is right-angled, and solves problems in two and three dimensions. Answers are often left as exact surds: a triangle with shorter sides 5 and 15 has hypotenuse \(\sqrt{250} = 5\sqrt{10}\).
What is a surd?
A surd is a root that cannot be written as a fraction, such as \(\sqrt{2}\) or \(\sqrt[3]{9}\). Its decimal never ends or repeats, so the surd itself is the exact value. \(\sqrt{324} = 18\) is not a surd, because 324 is a perfect square.
How do you simplify a surd?
Write the number under the root as a perfect square times another factor, then take the square root of the perfect square. For example, \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\).
Can you add any two surds?
Only like surds can be added. \(\sqrt{2} + \sqrt{8}\) is not \(\sqrt{10}\): simplify first, \(\sqrt{8} = 2\sqrt{2}\), so the sum is \(3\sqrt{2}\).
Which NESA outcomes does this topic cover?
Pythagoras' theorem is Stage 4 content under MA4‑PYT‑C‑01, revised in Year 9 Advanced. Surds and irrational numbers come from the Stage 5 Path outcome MA5‑IND‑P‑02, and the problems in solids look ahead to MA5‑TRG‑P‑01.