Question

geometry
directions: for the extension activities, you must complete a total of 12 points. each question has a number of points assigned to it. your teacher may assign you specific questions to do, or they may let you to choose.

1. find the distance between (2,0) and (6,0). (1 pt)
2. find the distance between (5,0) and (10,0). (1 pt)
3. find the distance between (2,4) and (6,12). (1 pt)
4. find the distance between (5,7) and (10,14). (1 pt)
5. find the distance between (2, - 4) and (-6,12). (1 pt)
6. find the distance between (-5,5) and (5, - 5). (1 pt)
7. find the fractional distance starting from (2,0) and going $\frac{3}{4}$ of the way to (6,0). (2 pt)
8. find the fractional distance starting from (6,0) and going $\frac{3}{4}$ of the way to (2,0). (2 pt)
9. find the fractional distance starting from (4,12) and going $\frac{1}{4}$ of the way to (16,3). (2 pt)
10. find the fractional distance starting from (16,3) and going $\frac{1}{4}$ of the way to (4,12). (2 pt)

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. When $y_1 = y_2$, the formula simplifies to $d=\vert x_2 - x_1\vert$.

Step2: Solve question 1

For points $(2,0)$ and $(6,0)$, since $y$-coordinates are the same, $d=\vert6 - 2\vert= 4$.

Step3: Solve question 2

For points $(5,0)$ and $(10,0)$, since $y$-coordinates are the same, $d=\vert10 - 5\vert = 5$.

Step4: Solve question 3

For points $(2,4)$ and $(6,12)$, use the full - distance formula: $d=\sqrt{(6 - 2)^2+(12 - 4)^2}=\sqrt{4^2+8^2}=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}$.

Step5: Solve question 4

For points $(5,7)$ and $(10,14)$, $d=\sqrt{(10 - 5)^2+(14 - 7)^2}=\sqrt{5^2+7^2}=\sqrt{25 + 49}=\sqrt{74}$.

Step6: Solve question 5

For points $(2,-4)$ and $(-6,12)$, $d=\sqrt{(-6 - 2)^2+(12+4)^2}=\sqrt{(-8)^2+16^2}=\sqrt{64 + 256}=\sqrt{320}=8\sqrt{5}$.

Step7: Solve question 6

For points $(-5,5)$ and $(5,-5)$, $d=\sqrt{(5 + 5)^2+(-5 - 5)^2}=\sqrt{10^2+(-10)^2}=\sqrt{100 + 100}=\sqrt{200}=10\sqrt{2}$.

Step8: Solve question 7

The $x$-coordinate of the point that is $\frac{3}{4}$ of the way from $(2,0)$ to $(6,0)$ is $x=2+\frac{3}{4}(6 - 2)=2 + 3=5$, so the point is $(5,0)$.

Step9: Solve question 8

The $x$-coordinate of the point that is $\frac{3}{4}$ of the way from $(6,0)$ to $(2,0)$ is $x=6-\frac{3}{4}(6 - 2)=6 - 3 = 3$, so the point is $(3,0)$.

Step10: Solve question 9

The $x$-coordinate of the point that is $\frac{1}{4}$ of the way from $(4,12)$ to $(16,3)$ is $x=4+\frac{1}{4}(16 - 4)=4 + 3=7$, and the $y$-coordinate is $y=12+\frac{1}{4}(3 - 12)=12-\frac{9}{4}=\frac{48 - 9}{4}=\frac{39}{4}$, so the point is $(7,\frac{39}{4})$.

Step11: Solve question 10

The $x$-coordinate of the point that is $\frac{1}{4}$ of the way from $(16,3)$ to $(4,12)$ is $x=16-\frac{1}{4}(16 - 4)=16 - 3 = 13$, and the $y$-coordinate is $y=3+\frac{1}{4}(12 - 3)=3+\frac{9}{4}=\frac{12 + 9}{4}=\frac{21}{4}$, so the point is $(13,\frac{21}{4})$.

Answer:

1. 4
2. 5
3. $4\sqrt{5}$
4. $\sqrt{74}$
5. $8\sqrt{5}$
6. $10\sqrt{2}$
7. $(5,0)$
8. $(3,0)$
9. $(7,\frac{39}{4})$
10. $(13,\frac{21}{4})$