Question

the area of a right angled triangle is 24 cm^2 . Given that its base is ( 2 - x) cm and its height is ( 4 - x ) cm. find the value of x and hence the length of the base and the height of the right.

Answer 1

x = -4

Explanation:

The area of a triangle is half the product of its base and height.

`\boxed{\begin{minipage}{5 cm}\underline{Area of a triangle}\\\\$A=(1)/(2)bh$\\\\where:\\ \phantom{ww}$\bullet$ $A$ is the area. \\ \phantom{ww}$\bullet$ $b$ is the base. \\ \phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}`

Given values:

• Area = 24 cm²
• Base = (2 - x) cm
• Height = (4 - x) cm

Substitute the given values into the formula:

`\implies 24=(1)/(2)(2-x)(4-x)`

Multiply both sides of the equation by 2:

`\implies 48=(2-x)(4-x)`

Expand the parentheses:

`\implies 48=8-2x-4x+x^2`

`\implies 48=8-6x+x^2`

Subtract 48 from both sides;

`\implies x^2-6x-40=0`

Rewrite -6x as -10x + 4x:

`\implies x^2-10x+4x-40=0`

Factor the first two terms and the last two terms separately:

`\implies x(x-10)+4(x-10)=0`

Factor out the common term (x - 10):

`\implies (x+4)(x-10)=0`

Apply the zero-product property:

`(x+4)=0 \implies x=-4`

`(x-10)=0 \implies x=10`

If x ≥ 4 then the height of the triangle would be h ≤ 0.

If x = 10, the height of the triangle would be -6.

As length cannot be negative, the only value of x is x = -4.