The equations of the three altitudes of triangle ABC include the following:
- 3y - 2y - 4 = 0.
- y + 3x - 8 = 0.
- 4y + x - 6 = 0.
A triangle can be defined as a two-dimensional geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.
<h3>What is a slope?</h3>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.
<h3>How to determine a slope?</h3>Mathematically, the slope of a straight line can be calculated by using this formula;
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:
- 3y - 2y - 4 = 0.
- y + 3x - 8 = 0.
- 4y + x - 6 = 0.
Read more on point-slope form here: brainly.com/question/24907633
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