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Rina draws a plan of her school on a coordinate grid. She plots the school gate at point p(-8,-1) and the library at point Q(2,5). She notices that the science lab is also located on PQ and is two thirds the distance from the library to the school gate. What are the coordinates of the science lab ?

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Ashraf82

Answer:

The coordinates of the science lab is:

x = -14/3  ,  y = 1

Step-by-step explanation:

∵ P is (-8 , -1) and Q is (2 , 5)

∵ The point of the science lab is (x , y)

∵ The science lab point divided PQ at ratio 2 : 1 from Q (two thirds PQ)

∴ [tex]x=\frac{m_{1}x_{p}+m_{2}x_{q}}{m_{1}+m_{2}}[/tex]

∴ [tex]y=\frac{m_{1}y_{p}+m_{2}y_{q}}{m_{1}+m_{2}}[/tex]

∴ [tex]x=\frac{2(-8)+1(2)}{1+2}=\frac{-14}{3}[/tex]

∴ [tex]y=\frac{2(-1)+1(5)}{2+1}=\frac{3}{3}=1[/tex]

∴ The point of science lab = (-14/3 , 1)

Using proportions, it is found that the coordinates of the science lab are [tex]\left(-\frac{4}{3}, 3\right)[/tex].

What is a proportion?

A proportion is a fraction of a total amount.

For the x-coordinate, we have that:

  • The x-coordinate of P is -8.
  • The x-coordinate of Q is 2.
  • The x-coordinate of the science lab is of x.
  • The science lab is two-thirds of the distance from Q to P, hence:

[tex]x - P = \frac{2}{3}(Q - P)[/tex]

[tex]x + 8 = \frac{2}{3}(10)[/tex]

[tex]x = \frac{20}{3} - \frac{24}{3} = -\frac{4}{3}[/tex]

Applying the same logic for the y-coordinate:

[tex]y - P = \frac{2}{3}(Q - P)[/tex]

[tex]y + 1 = \frac{2}{3}(6)[/tex]

[tex]y = 3[/tex]

The coordinates of the science lab are [tex]\left(-\frac{4}{3}, 3\right)[/tex].

More can be learned about proportions at https://brainly.com/question/24372153

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