SOLUTION: the verticed of a triangle abc are a(1,7) b(9,3) and c(3,1) a. prove that triangle ABC is a right triangle b.which angle is the right angle? c.which side is the hypotenu

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Question 566528: the verticed of a triangle abc are a(1,7) b(9,3) and c(3,1)
a. prove that triangle ABC is a right triangle

b.which angle is the right angle?
c.which side is the hypotenuse?
d.what are the coordinates of the midpoint of the hypotenuse?
e. what is the equation of the median from the vertex of the right angle to the hypotenuse?
f. what is the equation of the altitude from the vertex of the tight angle to the hypotenuse?

Please help ASAP. Thank you
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

The triangle is a right triangle if one of the angles is a right angle. If one of the angles is a right angle then two of the sides will be perpendicular. If two lines are perpendicular, then their slopes will be negative reciprocals of each other. That is to say that if one slope is then the slope of the perpendicular will be .

So your first step is to use the slope formula:

where and are the coordinates of the given points to find the slopes of the three lines that contain the segments that form the sides of the triangle.

Once you have the three slope numbers, determine if any pair of them are negative reciprocals of each other.

The point of intersection of the two sides that are perpendicular is the right angle vertex.

The hypotenuse is the side opposite the right angle, hence the endpoints of the hypotenuse are the two points that are NOT the right angle. Use the midpoint formulas to calculate the coordinates of the midpoint of the hypotenuse:

and

You can't write the equation of a median. The median is a line SEGMENT. You can write the equation of the line containing the median. The median from a vertex is the line segment that joins the vertex to the midpoint of the opposite side. You know the coordinates of the vertex from part b of this problem and you know the coordinates of the midpoint of the hypotenuse from part d of this question, so you know the coordinates of two points on the line containing the desired median segment. Use the two-point form of an equation of a line to derive the desired equation.

where and are the coordinates of the given points.

Simplify as necessary/desired.

The altitude from the vertex of the right angle to the hypotenuse is, by definition, perpendicular to the hypotenuse. Recall that perpendiculars have negative reciprocal slopes. You have the slope of the hypotenuse from the calculations you did for part a of this problem. Calculate the negative reciprocal of the slope of the hypotenuse to find the slope of any line perpendicular to the line containing the hypotenuse.

Then use the point-slope form of an equation of a line:

where are the coordinates of the right angle vertex (part b of this problem) and is the calculated slope.

Simplify as desired or required.

John

My calculator said it, I believe it, that settles it