9514 1404 393
(a) C(0, 0)
(b) D(4 2/3, 2 1/3)
(c) E(1.2, 0.6)
Step-by-step explanation:
All of these problems are similar, so let's talk about how to work them in general. The points to be found are C, D, E. For this discussion, let's say the point to be found is P.
"Similar triangles" means the triangle side lengths are proportional. When P is located some fraction from A to B, it means the horizontal triangle leg from Ax to Px is that fraction of the horizontal triangle leg from Ax to Bx. In equation form, it means ...
(Px -Ax)/(Bx -Ax) = fraction
The same is true for the vertical triangle legs:
(Py -Ay)/(By -Ay) = fraction
These equations are used to find the values of Px and Py, so the coordinates of P: (Px, Py).
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(a) (Cx -(-2))/(6 -(-2)) = 1/4 . . . . . . . substituting known values in the above equation
Cx +2 = 8(1/4) = 2 . . . . . multiply by 8
Cx = 0 . . . . . . . . . . . . . . add -2
The y-coordinate is found the same way.
(Cy -(-1))/(3 -(-1)) = 1/4
Cy +1 = 1 . . . . . . . . . . . . . multiply by 4
Cy = 0
The point 1/4 of the way from A to B is C(0, 0).
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(b) For this, the starting and ending points are switched, so the reference point is B, not A.
(Dx -Bx)/(Ax -Bx) = 1/6
(Dx -6)/(-2 -6) = 1/6
Dx -6 = -8/6 . . . . . . . . . . . multiply by -8
Dx = 6 -4/3 = 4 2/3 . . . . . add 6
The y-coordinate is found the same way.
(Dy -3)/(-1 -3) = 1/6
Dy -3 = -4/6 . . . . . . . . . multiply by -4
Dy = 2 1/3 . . . . . . . . . . . add 3
The point 1/6 of the way from B to A is D(4 2/3, 2 1/3).
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(c) We assume a typo in the problem statement, so that the desired ratio is ...
AE : EB = 2 : 3
This means our point E is located 2/5 of the way from A to B.
(Ex +2)/8 = 2/5 . . . . . . using previous results, we don't have to write out and compute the difference Bx -Ax
Ex = 8(2/5) -2 = 1 1/5
And the y-coordinate is computed similarly.
(Ey +1)/4 = 2/5
Ey = 8/5 -1 = 3/5
The point E dividing the segment AB into parts in the ratio 2:3 is E(1.2, 0.6).
