The diagram shows a right-angled triangle - Edexcel - GCSE Maths - Question 17 - 2019 - Paper 1

Expanding the equation gives:

$x^2 - 2x + 4x - 8 = 55$

Combining like terms results in:

$x^2 + 2x - 8 = 55$

Rearranging the equation leads to:

$x^2 + 2x - 63 = 0$

Next, we use the quadratic formula to solve for $x$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = 2$, and $c = -63$:

$x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-63)}}{2(1)}$

Calculating inside the square root gives:

$x = \frac{-2 \pm \sqrt{4 + 252}}{2}$

Which simplifies to:

$x = \frac{-2 \pm \sqrt{256}}{2}$

The roots are:

$x = \frac{-2 + 16}{2} = 7$

Or:

$x = \frac{-2 - 16}{2} = -9$

Since $x$ must be positive, we have $x = 7$.

Now calculating the lengths of the sides:

• Shortest side: $x - 2 = 7 - 2 = 5$ cm
• Base: $x + 4 = 7 + 4 = 11$ cm

Therefore, the length of the shortest side of the triangle is 5 cm.