Grade 8 Pythagoras Quiz 1 draws its questions at random from 21 questions across these topics:
A few of the question types, with full solutions, so you know what to expect.
The Theorem — Naming Sides. Triangle PQR is right-angled at Q (see diagram). Which side is the hypotenuse?
Answer: Side \(r\) (PR)
The hypotenuse is always the side opposite the right angle.
The right angle is at Q, so the hypotenuse is the side that does not touch Q — that is side PR, labelled \(r\).
It is also the longest side of the triangle.
The Theorem — The Formula. In a right-angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\), which statement is Pythagoras' Theorem?
Answer: \(c^2 = a^2 + b^2\)
Pythagoras' Theorem: the square of the hypotenuse equals the sum of the squares of the other two sides.
\[ c^2 = a^2 + b^2 \]
The Theorem — Areas of Squares. Squares are drawn on all three sides of a right-angled triangle (see diagram). The squares on the two legs have areas 9 cm\(^2\) and 16 cm\(^2\). What is the area of the square on the hypotenuse?
Answer: 25 cm\(^2\)
Pythagoras' Theorem is really a statement about areas:
\[ c^2 = a^2 + b^2 \]
Area on hypotenuse \(= 9 + 16 = 25\) cm\(^2\)
Finding the Hypotenuse. A right-angled triangle has legs of 3 cm and 4 cm (see diagram). Find the length of the hypotenuse \(x\).
Answer: 5 cm
\( x^2 = 3^2 + 4^2 \)
\( x^2 = 9 + 16 = 25 \)
\( x = \sqrt{25} = 5 \)
Finding the Hypotenuse. A right-angled triangle has legs of 5 cm and 12 cm (see diagram). Find the hypotenuse \(x\).
Answer: 13 cm
\( x^2 = 5^2 + 12^2 \)
\( x^2 = 25 + 144 = 169 \)
\( x = \sqrt{169} = 13 \)