Answer: (-6.5, -5)
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Step-by-step explanation:
Refer to the diagram below.
Let's define these two point labels
A = (-9, -10)
B = (-5, -2)
which are the two given endpoints of the segment. Let's add in point C such that
C = (-5, -10)
Notice triangle ABC is a right triangle. The 90 degree angle is at point C.
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The x coordinates of points A and C are -9 and -5 respectively. This is a gap of 4 units. Split this into 8 pieces to find each is 4/8 = 0.5 units wide. Why 8? Because the "ratio 5 to 3" has a total of 5+3 = 8 parts.
Since the first number of that ratio is 5, we'll move 5 parts to the right of point A. So we'll move 5*0.5 = 2.5 units to the right of A.
We go from A(-9,-10) to D(-6.5, -10)
Take note how the segments AD and DC are in ratio 5:3
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The y coordinates of B and C are -2 and -10 in that order. This is a gap of 8 units. Split this into 8 equal parts, and each part is 1 unit tall.
Start at point C and move upward 5 of those parts, ie 5*1 = 5 units up. You should arrive at point E(-5, -5) shown in the diagram below.
Then draw a vertical line upward from point D until arriving at segment AB. At the same time, draw a horizontal line from point E until arriving at segment AB. Both approaches should get to point (-6.5, -5) which is the final answer. This is denoted as point F in the diagram below (marked in red).