A point b on a segment with endpoints a (1,4) and c (8,2) partitions the segment in a 3:1 ratio. find the coordinates of point b
1 answer:
Our overall goal in this problem is to find the location of a point b that lies on the line between points A and C.
What do know from what's given?
Location of point A
Location of point C
The relative distance of point B along the segment/line AC (given as a ratio of 3:1)
Now that we have listed everything we know, lets figure out what we can calculate based on this information:
Since we have both the locations of points A and C, lets calculate the length of this segment. For this we can think of this edge as the hypotenuse of a right triangle that is formed from the height (which is the distance between the x-coordinates) and the base (which is the distance between the y-coordinates).
We'll need two equations here:
Distance:
Calculate the distances for the height and the base of this triangle formed from the segment AC (the hypotenuse of the right triangle)
a (base) = |x2 - x1| ... where A (x1,y1) and C (x2,y2)
b (height) = |y2 - y1| ... where A (x1,y1) and C (x2,y2)
Note: Absolute value is used here since distance is always positive
Pythagorean Theorem:
Once we know the distance of the base and the height, we'll be able to calculate the segment (aka the hypotenuse of this formed triangle)
c2 = a2 + b2
After we calculate the distances for a and b, we can then solve for c, giving us the length of segment AC. Great! We're halfway there - next is to figure out how far point B is from points A and C, using the distance along the segment AC.
Recall the third piece of information we were given: the relative distance of point T along segment AC. How can we represent this information in an equation using the distance of segment AC?
The ratio to which point T splits the segment AC is given as 3:1 - this means that one part is 3 times as long as the other part. The natural question to ask here is "3 times as long as what?" This 'what' piece is what we want to find out - lets call it x.
Another important observation to make is that this ratio is describing two parts of a whole segment - so if we think of this is two pieces of a whole, we can think of it as '3 times something' and '1 times something' is equivalent to the entire segment.
So, using this terminology with x, lets rewrite what we said in words into an equation:
3x + 1x = c (Length of AC)
Solving for x, we see that x =
Now that we have the distance from point D to point T, and from Point T to point F, we can take up the final step of determining the location of T.
Lets imagine the entire segment ABC as the hypotenuse of a triangle again. This time we want to imagine (feel free to draw this yourself as well) a like straight down from T, perpendicular to the base of this right triangle.
What we have now is a small right triangle (with segment DT as the hypotenuse) within a larger right triangle (with segment DF as the hypotenuse). What this means is that we have two proportional triangles.
With this information, we can create a proportion statement that says:
(AT) Hypotenuse small Triangle / Height smallTriangle ~ (AC) Hypotenuse large Triangle / (b) Height large Triangle
Recall that we had solved for b in the beginning as the y distance between points D and F. This means the only variable missing is Height small Triangle, which results to be:
Height small Triangle = (AT) Hypotenuse small Triangle × (b) Height large Triangle / (DF) Hypotenuse large Triangle
Adding Height small Triangle to the y coordinate of point F gives us point T's y coordinate.
Now that we have the hypotenuse length and the height length of the small triangle formed by AT (as the hypotenuse) we can plug in those numbers into the Pythagorean theorem to since for there final missing side: the base:
Base small Triangle 2 = (AT) Hypotenuse small Triangle 2 Height small Triangle 2
Finally, we add Base small Triangle to the x coordinate of point B gives us point T's x coordinate.
Learn more about point segment :
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