Respuesta :
The slopes are -5/3, 0, and undefined. Since 0 and undefined are opposite reciprocal slopes, they are perpendicular and therefore form a 90 degree angle making the triangle a right triangle.
The triangle is a right triangle with coordinates (1,6) (4,1) (1,1) .
What is the slope of a line ?
The slope of a line is the measure of the steepness and the direction of the line. The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line.
The formula of the slope of a line:
Slope = m = tan θ = (y2 - y1)/(x2 - x1)
What is slope of perpendicular lines?
A set of perpendicular lines always has 90º angle between them. Let us suppose we have two perpendicular lines l1 and l2 in the coordinate plane, inclined at angle θ1 and θ2 respectively with the x-axis, such that the given angles follow the external angle theorem as, θ2 = θ1 + 90º.
Therefore, their slopes can be given as,
m1 = tan θ1
m2 = tan (θ1 + 90º) = - cot θ1
⇒ m1 × m2 = -1
Thus, the product of slopes of two perpendicular lines is equal to -1.
According to the question
The slopes of the sides of the triangle
the triangle is a right triangle
Let vertex of right triangle be ABC
with coordinates
A = (1,6)
B = (4,1)
C = (1,1)
Now ,
Slope of line AB
m1 = [tex]\frac{1-6}{4-1}[/tex]
m1 = [tex]\frac{-5}{3}[/tex]
Slope of line BC
m2 = [tex]\frac{1-1}{1-4}[/tex]
m2 = 0
i.e it is a horizontal line
Slope of line AC
m3 = [tex]\frac{1-6}{1-1}[/tex]
m3 = undefined
i.e it is a vertical line
Therefore,
Horizontal and vertical lines are always perpendicular: therefore, two lines, one of which has a zero slope and the other an undefined slope are perpendicular.
Hence , the triangle is a right triangle with coordinates (1,6) (4,1) (1,1) .
To know more about slope of a line and perpendicular lines here:
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