If the ratio of the length of segment AC to the length of segment CB is 3:1, what is the y-coordinate of point C?

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15-04-2020
Mathematics

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If the ratio of the length of segment AC to the length of segment CB is 3:1, what is the y-coordinate of point C?

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calculista calculista · Answer 1 · 2020-04-20T15:34:32+02:00

Answer:

The coordinates of point C are (8,8.5)

Step-by-step explanation:

The picture of the question in the attached figure

Let

[tex](C_x,C_y)[/tex] ----> coordinates of point C

we have that

The horizontal distance AB is equal to

[tex]AB_x=10-2=8\ units[/tex]

The vertical distance AB is equal to

[tex]AB_y=10-4=6\ units[/tex]

Find the horizontal coordinate of point C

we know that

[tex]\frac{AC}{CB}=\frac{3}{1}[/tex]

so

[tex]\frac{AC_x}{CB_x}=\frac{3}{1}[/tex]

[tex]AC_x=3CB_x[/tex]----> equation A

[tex]AC_x+CB_x=8[/tex] ----> equation B

substitute equation A in equation B

[tex]3CB_x+CB_x=8[/tex]

[tex]4CB_x=8\\CB_x=2[/tex]

[tex]AC_x=3(2)=6[/tex]

so

The x-coordinate of point C is equal to the x-coordinate of point A plus the horizontal distance between the point A and point C

[tex]C_x=A_x+AC_x=2+6=8[/tex]

Find the vertical coordinate of point C

we know that

[tex]\frac{AC}{CB}=\frac{3}{1}[/tex]

so

[tex]\frac{AC_y}{CB_y}=\frac{3}{1}[/tex]

[tex]AC_y=3CB_y[/tex]----> equation A

[tex]AC_y+CB_y=6[/tex] ----> equation B

substitute equation A in equation B

[tex]3CB_y+CB_y=6[/tex]

[tex]4CB_y=6\\CB_y=1.5[/tex]

[tex]AC_y=3(1.5)=4.5[/tex]

so

The y-coordinate of point C is equal to the y-coordinate of point A plus the vertical distance between the point A and point C

[tex]C_y=A_y+AC_y=4+4.5=8.5[/tex]

therefore

The coordinates of point C are (8,8.5)

akposevictor akposevictor · Answer 2 · 2022-01-07T13:29:26+01:00

Using the internal section formula, the y-coordinate of point C is: 8.5.

Recall the Internal Section Formula

The internal Section formula used to determine the coordinates of a point that partitions a line segment is:

[tex]x = \frac{mx_2 + nx_1}{m + n} \\\\y = \frac{my_2 + ny_1}{m + n} \\\\[/tex]

Given:

A(2, 4) and B(10, 10)

3:1 = AC/CB = 3/1

Applying the internal division formula, let:

[tex]A(2, 4) = (x_1, y_1)\\\\B(10, 10) = (x_2, y_2)[/tex]

m = 3

n = 1

Plug in the values:

[tex]x = \frac{(3 \times 10) + (1 \times 2)}{3 + 1}\\\\x = \frac{30 + 2}{4} = \frac{32}{4} \\\\\mathbf{x = 8}[/tex]

[tex]y = \frac{(3 \times 10) + (1 \times 4)}{3 + 1} \\\\y = \frac{(30 + 4)}{4} \\\\\mathbf{y = 8.5}[/tex]

Therefore, using the internal section formula, the y-coordinate of point C is: 8.5.

Learn more about the internal section formula on:

https://brainly.com/question/9404701