The points A(1,5) and B(9,3) are part of a triangle ΔABC. The triangle has a right angle at A and and sides satisfy AB=2AC. Find a point C such that C lies above the line AB.
see related questions below
what is length AB?
sqrt(4+ 64) = sqrt 68 = 2 sqrt 17
so
length AC = sqrt 17
what is slope AC?
It is -1/slope AB
slope AB = -2/8 = -1/4
so
slope AC = 4
now point C (x,y)
Oh, wow, problem over
go right one from A
go up 4 from there
what is hypotenuse?
LOL sqrt 17 !!!!
so
x = 1 + 1 = 2
y = 5 + 4 = 9
C(2,9)
To find the point C that lies above the line AB, we need to determine the coordinates of the point C.
Given that triangle ABC has a right angle at A and AB = 2AC, we can start by finding the equation of the line AB.
1. Calculate the slope of the line AB:
The slope (m) is given by the formula: m = (y2 - y1) / (x2 - x1)
For A(1, 5) and B(9, 3), the slope of AB is: mAB = (3 - 5) / (9 - 1) = -2 / 8 = -1/4
2. Find the midpoint of AB:
The midpoint (M) is found by averaging the x-coordinates and the y-coordinates of A and B.
For A(1, 5) and B(9, 3), the midpoint is: M = ( (1 + 9) / 2, (5 + 3) / 2 ) = (5, 4)
3. Find the negative reciprocal of the slope AB:
The negative reciprocal of a line with slope m is -1/m.
For AB with slope mAB = -1/4, the negative reciprocal is: -1 / (-1/4) = 4
4. Find the equation of the line perpendicular to AB and passing through point M:
Using the point-slope form of a line, the equation of the line is: y - y1 = mPerpendicular(x - x1)
Given the point M(5, 4) and the slope mPerpendicular = 4, the equation becomes: y - 4 = 4(x - 5)
Simplifying the equation: y - 4 = 4x - 20
5. Determine the coordinates of point C:
Since C lies above line AB, the y-coordinate of point C must be greater than the y-coordinate of B(9, 3).
Thus, we can set the y-coordinate of C to a value greater than 3, while substituting it into the equation of the line from step 4.
Let's choose y = 6 (greater than 3) and solve for x:
6 - 4 = 4x - 20
2 = 4x - 20
4x = 2 + 20
4x = 22
x = 22 / 4 = 11/2
Therefore, the coordinates of point C are: C(11/2, 6).