Question

Given the points M(-3,-4) and T(5,0).Find the coordinates of the point Q on directed line segment MT that partitions MT in the ratio 2:3

Answer 1

The coordinates of point Q are $Q = \left(\frac{9}{5}, -\frac{8}{5} \right)$.

Let be M and T two Endpoints of a Line Segment divided at a Ratio a : b. By Geometry and Linear Algebra we know that the location of the Point Q by means of the following expression:

$MQ + QT = MT$ (1)

$MQ = \frac{a}{a+b}\cdot MT$ (2)

$QT = \frac{b}{a+b}\cdot MT$ (3)

By (3), we have:

$(T-Q) = \frac{a}{a+b}\cdot (T-M)$

$Q = T-\frac{a}{a+b}\cdot (T-M)$

If we know that $M(x,y) = (-3,-4)$, $T(x,y) = (5,0)$, $a = 2$ and $b = 3$, then the coordinates of point Q that partitions MT in the ratio 2 : 3 is:

$Q = (5,0) -\frac{2}{5}\cdot [(5,0)-(-3,-4)]$

$Q = (5,0) -\frac{2}{5}\cdot (8,4)$

$Q = \left(\frac{9}{5}, -\frac{8}{5} \right)$

The coordinates of point Q are $Q = \left(\frac{9}{5}, -\frac{8}{5} \right)$.

Please see this question related to Segment Partition: https://brainly.com/question/3148758

Answer 2

Using line segments, it is found that the point is $\mathbf{Q(\frac{1}{5}, -\frac{8}{5})}$

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• Point M is located at (-3, -4).
• Point T is located at (5,0).
• Point Q is located at (x,y).
• Point Q partitions segment MT in the ratio 2/3, thus:

$Q - M = \frac{2}{5}(T - M)$

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• The x-coordinate of M is -3.
• The x-coordinate of T is 5.
• The x-coordinate of Q is x.

$Q - M = \frac{2}{5}(T - M)$

$x - (-3) = \frac{2}{5}(5 - (-3))$

$x + 3 = \frac{16}{5}$

$x = \frac{16}{5} - \frac{15}{5}$

$x = \frac{1}{5}$

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• The y-coordinate of M is -4.
• The y-coordinate of T is 0.
• The y-coordinate of Q is y.

$Q - M = \frac{2}{5}(T - M)$

$y - (-4) = \frac{2}{5}(0 - (-4))$

$y + 4 = \frac{8}{5}$

$y = \frac{8}{5} - \frac{16}{5}$

$y = -\frac{8}{5}$

The coordinates are $\mathbf{Q(\frac{1}{5}, -\frac{8}{5})}$

A similar problem is given at https://brainly.com/question/4871016