Question

ext 1.01 - distance/fractional distance
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 3/4 of the way to (6,0). (2 pt)
8. find the fractional distance starting from (6,0) and going 3/4 of the way to (2,0). (2 pt)
9. find the fractional distance starting from (4,12) and going 1/4 of the way to (16,3). (2 pt)
10. find the fractional distance starting from (16,3) and going 1/4 of the way to (4,12). (2 pt)

Explanation:

1.

Step1: Identify distance formula

For two - dimensional points \((x_1,y_1)\) and \((x_2,y_2)\), the distance \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 2,y_1=0,x_2 = 6,y_2 = 0\).
\[d=\sqrt{(6 - 2)^2+(0 - 0)^2}\]

Step2: Simplify the expression

\[d=\sqrt{4^2+0^2}=\sqrt{16}=4\]

Step1: Apply distance formula

For points \((x_1,y_1)=(5,0)\) and \((x_2,y_2)=(10,0)\), \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
\[d=\sqrt{(10 - 5)^2+(0 - 0)^2}\]

Step2: Simplify

\[d=\sqrt{5^2+0^2}=\sqrt{25}=5\]

Step1: Use distance formula

For \((x_1,y_1)=(2,4)\) and \((x_2,y_2)=(6,12)\), \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
\[d=\sqrt{(6 - 2)^2+(12 - 4)^2}\]

Step2: Calculate values inside square - root

\((6 - 2)^2=4^2 = 16\) and \((12 - 4)^2=8^2 = 64\). Then \(d=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}\)

Answer:

4
2.