Question

The height of a triangle is 8 m more than twice the length of the base. The area of the triangle is 21 m². Find the height of the triangle.

Answer 1

The height of the triangle is 14 m.

Explanation:

The area of a triangle can be found by using the following formula:

`\boxed{A=(1)/(2)bh}`

where:

• b is the base of the triangle.
• h is the height of the triangle.

If the height of a triangle is 8 m more than twice the length of the base, then an expression for h in terms of b is:

`h = 2b + 8`

Rewrite this equation to make b the subject:

`\implies 2b=h-8`

`\implies b=(1)/(2)h-4`

Given the area of the triangle is 21 m², substitute this value along with the found expression for b into the formula for area of a triangle:

`\implies (1)/(2)\left((1)/(2)h-4\right)h=21`

To find the height of the triangle, solve the equation for h.

`\implies (1)/(4)h^2-2h=21`

`\implies h^2-8h=84`

`\implies h^2-8h-84=0`

`\implies h^2-14h+6h-84=0`

`\implies h(h-14)+6(h-14)=0`

`\implies (h+6)(h-14)=0`

Apply the zero-product property:

`h+6=0 \implies h=-6`

`h-14=0 \implies h=14`

As length cannot be negative, the height of the triangle is 14 m.

Answer 2

The height of the triangle is 14 m,

Explanation:

Let the base be x, then the height is:

• 2x + 8

The area is half the product of base and height.

Set up equation and solve for x:

• x(2x + 8)/2 = 21
• x(x + 4) = 21
• x² + 4x = 21
• x² + 4x - 21 = 0
• x² + 7x - 3x - 21 = 0
• x(x + 7) - 3(x + 7) = 0
• (x + 7)(x - 3) = 0
• x = - 7 or x = 3

The first root is discarded as negative. The second root is the answer.

The base is 3 m, so the height is:

• 2*3 + 8 = 14 m