<p>Test 1 #42
Point Y lies on line segment XZ, between X and Z. The (x,y) coordinates for X and Z are (7,2) and (-3,-3), respectively. If the ratio of XY to YZ is 2 to 3, what are the (x,y) coordinates of Y?</p>
<p>F. (10, 5)
G. (4, -1)
H. (3, 2)
J. (3, 0)
K (2, 3)</p>
<p>Answer is supposed to be J, but I don't see how that could be.. G would make it a 3:2 ratio and the other ones don't make any sense. Is this a mistake?</p>
<p>Hmm I got J as the answer</p>
<p>How did you guys go about solving it?</p>
<p>First I found the distance of XZ. Then I found 2/5 of XZ. Then used that answer in the distance formula. So it ends up:
20=(X-7)squared + (Y-2)squared. Then I plugged in numbers. </p>
<p>20=(3-7)squared+(0-2)squared
20=16 + 4
20=20</p>
<p>At least I think...I definitely could be wrong. I'm not that great at math.</p>
<p>As you go from X to Z, the x coordinate goes 7 -> -3 (a decrease of 10), and the y-coordinate goes 2 -> -3 (a decrease of 5). </p>
<p>XY:YZ is 2:3, so XY is 2/(2+3) or 40% of the length of XZ. The x- and y-coordinates will drop by (40%)(10) and (40%)(5) respectively, as you move from X to Y; so the coordinates at Y will be</p>
<p>(x-coord of X - (0.4)(10), y-coord of X - (0.4)(5))
= ( 7 - 4, 2 - 2)
= (3, 0)</p>
<p>
[quote]
XY:YZ is 2:3, so XY is 2/(2+3) or 40% of the length of XZ.
[/quote]
</p>
<p>How did you get the (2+3) in 2/(2+3)?</p>
<p>Thanks</p>
<p>Jai6638:
XZ = XY + YZ
Since XY:YZ is 2:3, XZ = XY + (3/2).XY = 2.5 XY
or, XY = (1/2.5) XZ = (2/5) XZ .</p>
<p>You can also see it as
XY / XZ = XY / (XY + YZ) = 2a / (2a + 3a) = 2/ (2+3)
where a is some arbitrary multiple; the actual value doesn't matter, it cancels out.</p>
<p>Or alternatively (to make things needlessly complicated): you can use X and Z to form a right triangle with a third vertex (the 90 degree one) at (7,-3). You just find this from the intersection of the lines that go through X and Z that are parallel to the X axis and the Y axis respectively (it doesn't matter which you choose for which, just keep it constant). Anyway, so then you find the length of XZ which is about 11.18. Then multiply that number by 2/5 (the ratio). You should get XY = 4.427 and YZ = 6.708. Now make an educated guess along the line XZ as to where Y is. Draw a line down to the base (I chose the long side, length 10) of the large triangle from Y. You should end up with two similar triangles. Therefore, you can use ratios to determine the length of the two legs of the triangle that includes point Y. 11.18/10 = 6.708/x cross multiply to find x=6. Looking at the ratios of the larger triangle you can see that the long side is twice the short side (10:5). Therefore, the other leg of the smaller triangle is 3. Add 3 and 6 to the coordinates of your starting point (Z in this case) to get (3,0). Alright, this seems very convoluted but it's easy to see if you draw it.</p>