what are the coordinates of the three points that divide the line segment joining P(a,b) and Q(c,d) into four equal parts ...textbook answer is (3a+c/4,3b+d/4) is one of the three coordinats
Use the point of division formula.
Let the divided segment be
PABCQ
Then
r1=PA/PQ=1/4
r2=PB/PQ=2/4
r3=PC/PQ=3/4
Applying the point of division formula to points A, B and C:
A(a+r1(c-a), b+r2(d-b))
B(a+r2(c-a), b+r2(d-b))
C(a+r3(c-a), b+r3(d-b))
For more explanations and examples, see:
http://www.mathwords.com/p/point_of_division_formula.htm
To find the coordinates of the three points that divide the line segment PQ into four equal parts, you can use the concept of section formula or the midpoint formula. Let's use the section formula in this case.
The section formula states that if you divide a line segment into parts in the ratio m:n, then the coordinates of the point dividing the line in such a ratio are given by:
[(n * x1 + m * x2) / (m + n), (n * y1 + m * y2) / (m + n)]
Now, let's apply this formula to divide the line segment joining P(a, b) and Q(c, d) into four equal parts (where m = 3 and n = 1).
The first point, let's call it R1, will divide the line segment into 3:1 ratio:
Rx = [(1 * a + 3 * c) / (1 + 3)]
Ry = [(1 * b + 3 * d) / (1 + 3)]
So, the coordinates of R1 will be (a + 3c)/4, (b + 3d)/4.
The second point, let's call it R2, will divide the line segment into 2:2 ratio:
Rx = [(2 * a + 2 * c) / (2 + 2)]
Ry = [(2 * b + 2 * d) / (2 + 2)]
So, the coordinates of R2 will be (2a + 2c)/4, (2b + 2d)/4 = (a + c)/2, (b + d)/2.
The third point, let's call it R3, will divide the line segment into 1:3 ratio:
Rx = [(3 * a + 1 * c) / (3 + 1)]
Ry = [(3 * b + 1 * d) / (3 + 1)]
So, the coordinates of R3 will be (3a + c)/4, (3b + d)/4.
Therefore, the three coordinates that divide the line segment PQ into four equal parts are:
[(a + 3c)/4, (b + 3d)/4]
[(a + c)/2, (b + d)/2]
[(3a + c)/4, (3b + d)/4]
Hope this explanation helps!