Exercise solution 2, page 482 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) In a right-angled triangle, one angle is exactly \(90^\circ\). The side opposite the right angle is called the hypotenuse. It is always the longest side of the triangle. The other two sides are the shorter sides that form the right angle.

B) Pythagoras’ theorem applies only to right-angled triangles. It says:

\[ \text{(hypotenuse)}^2 = \text{(first shorter side)}^2 + \text{(second shorter side)}^2 \]

For example, if the hypotenuse is \(m\) and the other sides are \(p\) and \(q\), then:

\[ m^2 = p^2 + q^2 \]

C) To write the correct statement, first identify the hypotenuse. A quick method is to look for the right-angle mark: the hypotenuse is the side directly opposite it. Once you know this side, its square must be alone on one side of the equation.

For example, if \(r\) is opposite the right angle, then the correct form is:

\[ r^2 = s^2 + t^2 \]

It would be incorrect to write \(s^2 = r^2 + t^2\), because the hypotenuse must be the side being squared by itself.

D) Squaring a length means multiplying it by itself. For example:

\[ 6^2 = 6 \times 6 = 36 \]

So in Pythagoras’ theorem, \(a^2\), \(b^2\) and \(c^2\) are not the side lengths themselves, but the squares of those lengths.

E) Sometimes the correct statement can also be rearranged. If

\[ u^2 = v^2 + w^2 \]

then you can subtract one squared term from both sides to get:

\[\begin{aligned} u^2 &= v^2 + w^2 \\ u^2 - v^2 &= w^2 \end{aligned}\]

So forms such as \(w^2 = u^2 - v^2\) are possible, but only if they come from a correct Pythagoras equation to begin with.

F) A useful check is that the hypotenuse is the longest side, so its square should be equal to the sum of the squares of the two shorter sides. If an equation says a shorter side squared equals the sum of the other two squares, then it cannot be correct for a right-angled triangle.

G) When choosing between several statements, do not guess from the letters. The letters can be placed on any sides. Always use the diagram: find the right angle, identify the opposite side, then write or select the equation in the form

\[ \text{hypotenuse}^2 = \text{side}^2 + \text{side}^2 \]