Question

directions: answer each question below. make sure you show all your work and round your answers to the nearest hundredth when necessary! 1) ef has a mid - point located at (2,9). if point e is located at (-12,19) and point f is located at (16,y), find the value of y. 2) △abc sits in the coordinate plane and has vertices (1,4),(-2,7), and (-1,-1) respectively. find the perimeter of this triangle. 3) circle the two sets of points that have the same mid - point. (-2,7) and (4,-1) (4,-4) and (8,0) (1,9) and (-4,0) (6,10) and (-4,-7) (-3,7) and (,2) 4) two trains leave the station at the same time. one is traveling west at a speed of 70mph. the other travels south at a speed of 85mph. after 6 hours, how far apart are the two trains? 5) find the second endpoint of the line segment that begins at point g(-2,5) and has a mid - point at m(1,7). 6) rank the distances between each of the following sets of points from shortest to longest. a) (-11,12) and (1,16) b) (-1,2) and (14,1) c) (6,9) and (7,-10) d) (2,3) and (10,-2) 7) in the standard (x,y) coordinate plane, line segment (overline{ab}) has endpoints (3,-4) and (-7,10). if (a,b) is the mid - point of (overline{ab}), then what is a + b? 8) calculate the distance between the following two points a(x,x + 3) and k(4x,x - 6)

Explanation:

1.

Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Given $E(-12,19)$, $F(16,y)$ and mid - point $(2,9)$. For the $x$ - coordinate of the mid - point: $\frac{-12 + 16}{2}=2$ (which is correct). For the $y$ - coordinate of the mid - point: $\frac{19 + y}{2}=9$.

Step2: Solve for $y$

Multiply both sides of the equation $\frac{19 + y}{2}=9$ by 2: $19 + y=18$. Then subtract 19 from both sides: $y=18 - 19=-1$.

Step1: Recall the distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Let $A(1,4)$, $B(-2,7)$ and $C(-1,-1)$.
First, find the distance between $A(1,4)$ and $B(-2,7)$:
$d_{AB}=\sqrt{(-2 - 1)^2+(7 - 4)^2}=\sqrt{(-3)^2+3^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}$.

Step2: Find the distance between $B(-2,7)$ and $C(-1,-1)$

$d_{BC}=\sqrt{(-1+2)^2+(-1 - 7)^2}=\sqrt{1^2+(-8)^2}=\sqrt{1 + 64}=\sqrt{65}$.

Step3: Find the distance between $C(-1,-1)$ and $A(1,4)$

$d_{CA}=\sqrt{(1 + 1)^2+(4 + 1)^2}=\sqrt{2^2+5^2}=\sqrt{4 + 25}=\sqrt{29}$.

Step4: Calculate the perimeter

$P=d_{AB}+d_{BC}+d_{CA}=3\sqrt{2}+\sqrt{65}+\sqrt{29}\approx3\times1.414+8.062+5.385=4.242+8.062+5.385 = 17.69$.

Step1: Apply mid - point formula for each pair

For the pair $(-2,7)$ and $(4,-1)$: Mid - point $=(\frac{-2 + 4}{2},\frac{7-1}{2})=(1,3)$.
For the pair $(4,-4)$ and $(8,0)$: Mid - point $=(\frac{4 + 8}{2},\frac{-4+0}{2})=(6,-2)$.
For the pair $(1,9)$ and $(-4,0)$: Mid - point $=(\frac{1-4}{2},\frac{9 + 0}{2})=(-\frac{3}{2},\frac{9}{2})$.
For the pair $(6,10)$ and $(-4,-7)$: Mid - point $=(\frac{6-4}{2},\frac{10-7}{2})=(1,\frac{3}{2})$.
For the pair $(-3,7)$ and $(?,2)$ (let the unknown $x$ - coordinate be $x$), mid - point $=(\frac{-3 + x}{2},\frac{7 + 2}{2})=(\frac{-3 + x}{2},\frac{9}{2})$.
The pairs $(-2,7)$ and $(4,-1)$ and $(6,10)$ and $(-4,-7)$ do not have the same mid - point. The pairs $(-2,7)$ and $(4,-1)$ and $(1,9)$ and $(-4,0)$ do not have the same mid - point. The pairs $(4,-4)$ and $(8,0)$ and $(1,9)$ and $(-4,0)$ do not have the same mid - point. The pairs $(4,-4)$ and $(8,0)$ and $(6,10)$ and $(-4,-7)$ do not have the same mid - point.
The pairs $(-2,7)$ and $(4,-1)$ and $(-3,7)$ and $(5,2)$ (since if $\frac{-3 + x}{2}=1$, then $x = 5$) have the mid - point $(1,3)$.

Answer:

$y=-1$

2.