Respuesta :

Answer:

A.[tex]\frac{RP}{SP}=3[/tex]

B.[tex]\frac{RP}{RS}=\frac{3}{4}[/tex]

Step-by-step explanation:

The complete question is

Three collinear points on the coordinate plane are R(x, y), S(x+8h, y+8k), and P(x+6h, y+6k).

Part A: Determine the value of RP/SP

Part B: Determine the value of RP/RS

we know that

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

we have

[tex]R(x,y),S(x+8h,y+8k) and P(x+6h,y+6k)[/tex]

Part A.We have to find the value of [tex]\frac{RP}{SP}[/tex]

step 1

Find the distance RP

[tex]R(x,y),P(x+6h,y+6k)[/tex]

substitute the values in the formula

[tex]RP=\sqrt{(x+6h-x)^2+(y+6k-y)^2}[/tex]

[tex]RP=\sqrt{36h^2+36 k^2}[/tex]

[tex]RP=6\sqrt{h^2+k^2}[/tex]

step 2

Find the distance SP

[tex]S(x+8h,y+8k),P(x+6h,y+6k)[/tex]

substitute the values in the formula

[tex]SP=\sqrt{(x+6h-x-8h)^2+(y+6k-y-8k)^2}[/tex]

[tex]SP=\sqrt{4h^2+4k^2}[/tex]

[tex]SP=\sqrt{4(h^2+k^2)}[/tex]

[tex]SP=2\sqrt{h^2+k^2}[/tex]

step 3

Find the ratio  RP/SP

[tex]\frac{RP}{SP}=\frac{6\sqrt{h^2+k^2}}{2\sqrt{h^2+k^2}}[/tex]

[tex]\frac{RP}{SP}=3[/tex]

Part B. We have to determine the value of [tex]\frac{RP}{RS}[/tex]

step 1

Find the distance RS

[tex]R(x,y),S(x+8h,y+8k)[/tex]

[tex]RS=\sqrt{(x+8h-x)^2+(y+8k-y)^2}[/tex]

[tex]RS=\sqrt{64h^2+64k^2}[/tex]

[tex]RS=\sqrt{64(h^2+k^2)}[/tex]

[tex]RS=8\sqrt{h^2+k^2}[/tex]

step 2

Find the ratio RP/RS

[tex]\frac{RP}{RS}=\frac{6\sqrt{h^2+k^2}}{8\sqrt{h^2+k^2}}[/tex]

[tex]\frac{RP}{RS}=\frac{3}{4}[/tex]

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