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?

Answer 1

To find the value of \( x \) for the triangle with sides \( x \), \( 2x \), and \( x^2 \) and a perimeter of 10 cm, we can start by expressing the perimeter in terms of \( x \).

The perimeter \( P \) of the triangle is given by:

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

Simplifying this equation, we have:

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

Rearranging the equation, we get:

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

Next, we will solve this quadratic equation using the quadratic formula:

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

where \( a = 1 \), \( b = 3 \), and \( c = -10 \).

Calculating the discriminant:

\[
b^2 - 4ac = 3^2 - 4(1)(-10) = 9 + 40 = 49
\]

Now substituting into the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{49}}{2 \cdot 1} = \frac{-3 \pm 7}{2}
\]

Calculating the two possible values for \( x \):

1. \( x = \frac{-3 + 7}{2} = \frac{4}{2} = 2 \)
2. \( x = \frac{-3 - 7}{2} = \frac{-10}{2} = -5 \) (not valid since \( x \) must be positive)

Thus, the only valid solution is:

\[
x = 2
\]

We can verify that the sides of the triangle are:
- \( x = 2 \)
- \( 2x = 4 \)
- \( x^2 = 4 \)

Now, we check the perimeter:

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

Since these values satisfy the perimeter and confirm that they fulfill the triangle inequality (each side must be less than the sum of the other two sides), we conclude that:

\[
\text{The value of } x \text{ is } \boxed{2}.
\]