the vertices of triangle abc are A(-2,3) B(10,3) and C(4,10. prove by coordinate geometry that the median to side AB is also the altitude to side BC?
To prove that the median to side AB is also the altitude to side BC, we need to show that they intersect at a right angle.
Let's start by finding the midpoint of side AB, which is the vertex C. We can use the midpoint formula:
Midpoint = [(x1 + x2)/2, (y1 + y2)/2]
Midpoint of AB = [(-2 + 10)/2, (3 + 3)/2] = [4, 3]
Next, let's find the slope of side AB using the formula:
Slope = (y2 - y1)/(x2 - x1)
Slope of AB = (3 - 3)/(10 - (-2)) = 0/12 = 0
Since the slope of AB is 0, the equation of the line containing AB is y = 3.
Now, let's find the equation of the line that passes through C and is perpendicular to AB. Since the median to side AB is perpendicular to AB, it must have a slope that is the negative reciprocal of AB's slope.
The slope of the perpendicular line = -1/0 (negative reciprocal of AB's slope)
As the slope is undefined (division by zero), we can say that the perpendicular line is a vertical line passing through the x-coordinate of C, which is 4. Therefore, the equation of the perpendicular line is x = 4.
Now, let's find the intersection point of the perpendicular line (altitude) and side BC.
The equation of side BC is given by: y = mx + b, where m is the slope and b is the y-intercept. To find the slope, we use the formula:
Slope of BC = (10 - 3)/(4 - 4) = 7/0 (undefined)
As the slope is undefined, side BC is a vertical line passing through the x-coordinate of C, which is 4. Therefore, the equation of side BC is x = 4.
Since the perpendicular line and side BC are both vertical lines and have the same x-coordinate of 4, they intersect at every point where x = 4. This means that they intersect at C (4, 10).
Therefore, we have shown that the median to side AB (which passes through the midpoint of AB [4, 3]) and the altitude to side BC (which passes through C [4, 10]) intersect at the point C. Since they both intersect at C and both lines are vertical, they intersect at a right angle.
Hence, we have proven that the median to side AB is also the altitude to side BC using coordinate geometry.