Question

A triangle has a base of (3x + 7) and a height of (5x - 1). A second

triangle is drawn with a base that is tripled and a height that is
doubled. Find the difference between the area of the original triangle
and the area of the new triangle.

Answer 1

The difference between the area of the original triangle and the area of the new triangle is
`\Delta A_(\bigtriangleup) = (5)/(2)\cdot (3\cdot x +7)\cdot (5\cdot x -1)`.

Explanation:

The equation for the area of a triangle (
`A_(\bigtriangleup)`) is:

`A_(\bigtriangleup) = (1)/(2)\cdot b \cdot h`

Where:

`b` - Base, dimensionless.

`h` - Height, dimensionless.

The expression for each triangle are described below:

First Triangle (
`b = 3\cdot x + 7`,
`h = 5\cdot x - 1`)

`A_(\bigtriangleup,1) = (1)/(2)\cdot (3\cdot x+7)\cdot (5\cdot x -1)`

Second Triangle (
`b = 3\cdot (3\cdot x+7)`,
`h = 2\cdot (5\cdot x -1)`)

`A_(\bigtriangleup,2) = 3\cdot (3\cdot x+7)\cdot (5\cdot x -1)`

The difference between the area of the original triangle and the area of the new triangle is:

`\Delta A_(\bigtriangleup) = A_(\bigtriangleup,2)-A_(\bigtriangleup,1)`

`\Delta A_(\bigtriangleup) = 3\cdot (3\cdot x+7)\cdot (5\cdot x-1)-(1)/(2) \cdot (3\cdot x+7)\cdot (5\cdot x-1)`

`\Delta A_(\bigtriangleup) = (5)/(2)\cdot (3\cdot x +7)\cdot (5\cdot x -1)`

The difference between the area of the original triangle and the area of the new triangle is
`\Delta A_(\bigtriangleup) = (5)/(2)\cdot (3\cdot x +7)\cdot (5\cdot x -1)`.