Question

The length of the sides of a triangle are x, 2x and x² respectively. The perimeter of the triangle is 10cm. What is the value of x in centimetres? Show simple working out.

Answer 1

Given the lengths of the sides of the triangle \( x \), \( 2x \), and \( x^2 \), we can write the equation for the perimeter:

\[
x + 2x + x^2 = 10
\]

Simplifying this, we have:

\[
3x + x^2 = 10
\]

Rearranging the equation gives us:

\[
x^2 + 3x - 10 = 0
\]

Now we can solve this quadratic equation using the quadratic formula, where \( a = 1 \), \( b = 3 \), and \( c = -10 \):

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

Substituting the values of \( a \), \( b \), and \( c \):

\[
x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1}
\]

Calculating the discriminant:

\[
x = \frac{-3 \pm \sqrt{9 + 40}}{2}
\]

\[
x = \frac{-3 \pm \sqrt{49}}{2}
\]

\[
x = \frac{-3 \pm 7}{2}
\]

This gives us two potential solutions:

1. \( x = \frac{4}{2} = 2 \)
2. \( x = \frac{-10}{2} = -5 \)

Since \( x \) must be a positive length, we take:

\[
x = 2
\]

Now, let's confirm that these values satisfy the triangle inequality and the perimeter:

1. The sides of the triangle are: \( x = 2 \), \( 2x = 4 \), and \( x^2 = 4 \).
2. The perimeter is:

\[
2 + 4 + 4 = 10 \text{ cm (correct)}
\]

3. Checking triangle inequality:

- \( 2 + 4 > 4 \) (true)
- \( 2 + 4 > 4 \) (true)
- \( 4 + 4 > 2 \) (true)

All conditions are satisfied, thus the value of \( x \) is:

\[
\boxed{2} \text{ cm}
\]