Question

△ABC has vertices A(-2, 0), B(0,8), and C(4,2) Find the equations of the three altitudes of △ABC

Answer 1

The equations of the three altitudes of triangle ABC include the following:

1. 3y - 2y - 4 = 0.
2. y + 3x - 8 = 0.
3. 4y + x - 6 = 0.

What is a triangle?

A triangle can be defined as a two-dimensional geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.

What is a slope?

A slope is also referred to as gradient and it's typically used to describe both the ratio, direction and steepness of the function of a straight line.

How to determine a slope?

Mathematically, the slope of a straight line can be calculated by using this formula;

$Slope, m = \frac{Change\;in\;y\;axis}{Change\;in\;x\;axis}\\\\Slope, m = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}$

Also, the point-slope form of a straight line is given by this equation:

y - y₁ = m(x - x₁)

Assuming the following parameters for triangle ABC:

• Let AM be the altitudes on BC.

• Let BN be the altitudes on CA.

• Let CL be the altitudes on AB.

For the equation of altitude AM, we have:

Slope of BC = (2 - 8)/(4 - 0)

Slope of BC = -6/4

Slope of BC = -3/2

Slope of AM = -1/slope of BC

Slope of AM = -1/(-3/2)

Slope of AM = 2/3.

The equation of altitude AM is given by:

y - y₁ = m(x - x₁)

y - 0 = 2/3(x - (-2))

3y = 2(x + 2)

3y = 2x + 4

3y - 2y - 4 = 0.

For the equation of altitude BN, we have:

Slope of CA = (2 - 0)/(4 - (-2))

Slope of CA = 2/6

Slope of CA = 1/3

Slope of BN = -1/slope of CA

Slope of BN = -1/(1/3)

Slope of BN = -3.

The equation of altitude BN is given by:

y - y₁ = m(x - x₁)

y - 8 = -3(x - 0)

y - 8 = -3x

y + 3x - 8 = 0.

For the equation of altitude CL, we have:

Slope of AB = (8 - 0)/(0 - (-2))

Slope of AB = 8/2

Slope of AB = 4

Slope of CL = -1/slope of AB

Slope of CL = -1/4

The equation of altitude CL is given by:

y - y₁ = m(x - x₁)

y - 2 = -1/4(x - 4)

4y - 2= -(x - 4)

4y - 2= -x + 4

4y + x - 2 - 4 = 0.

4y + x - 6 = 0.

In conclusion, we can infer and logically deduce that the equations of the three altitudes of triangle ABC include the following:

1. 3y - 2y - 4 = 0.
2. y + 3x - 8 = 0.
3. 4y + x - 6 = 0.

Read more on point-slope form here: brainly.com/question/24907633

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