Math

QuestionFind the coordinates that divide the line segment from (2,6)(2,-6) to (5,6)(5,6) in a 1:5 ratio.

Studdy Solution

STEP 1

Assumptions1. The starting point of the directed line segment is (,6)(,-6). The ending point of the directed line segment is (5,6)(5,6)3. The line segment is partitioned in a ratio of1 to5

STEP 2

We can find the coordinates of the point that partitions the line segment in a given ratio using the section formula. The section formula in vector form is given by(x,y)=(mx2+nx1m+n,my2+ny1m+n)(x,y) = \left( \frac{m \cdot x2 + n \cdot x1}{m + n}, \frac{m \cdot y2 + n \cdot y1}{m + n} \right)where (x1,y1)(x1, y1) and (x2,y2)(x2, y2) are the coordinates of the two points, and m:nm:n is the given ratio.

STEP 3

Now, plug in the given values for (x1,y1)(x1, y1), (x2,y2)(x2, y2), and the ratio m:nm:n into the section formula.
(x,y)=(15+521+5,16+5(6)1+5)(x,y) = \left( \frac{1 \cdot5 +5 \cdot2}{1 +5}, \frac{1 \cdot6 +5 \cdot (-6)}{1 +5} \right)

STEP 4

implify the expressions in the numerator and denominator of each fraction.
(x,y)=(+106,6306)(x,y) = \left( \frac{ +10}{6}, \frac{6 -30}{6} \right)

STEP 5

Calculate the values of xx and yy.
(x,y)=(15,24)(x,y) = \left( \frac{15}{}, \frac{-24}{} \right)

STEP 6

implify the fractions to get the final coordinates of the point.
(x,y)=(2.5,4)(x,y) = \left(2.5, -4 \right)The coordinates of the point on the directed line segment from (2,6)(2,-6) to (5,6)(5,6) that partitions the segment into a ratio of1 to5 are (2.5,4)(2.5, -4).

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