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# Alice is on the line x=4, and she is √40 away from (2,3). What is the sum of the possible y-coordinates of the point Alice is at?

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Alice is on the line x=4, and she is  √40 away from (2,3). What is the sum of the possible y-coordinates of the point Alice is at?

Aug 19, 2018
edited by Guest  Aug 19, 2018

#1
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Alice is one the line  x = 4  ,  so the  x-coordinate of whatever point Alice is on is  4 .

Let's call the point that Alice is located  (4, y)  .

By the distance forumula...

the distance between  (4, y)  and  (2, 3)   =   $$\sqrt{(4-2)^2+(y-3)^2}$$

The problem tells us that the distance between  (4, y)  and  (2, 3)  is  √40 ,  so...

$$\sqrt{(4-2)^2+(y-3)^2}=\sqrt{40}$$

Now let's solve this equation for  y .  First square both sides.

$$(4-2)^2+(y-3)^2=40$$

Simplify  $$(4-2)^2$$  to  $$4$$ .

$$4+(y-3)^2=40$$

Subtract  4  from both sides of the equation.

$$(y-3)^2=36$$

Take the ± square root of both sides.

$$y-3=\pm\sqrt{36}$$

$$y-3=\pm6$$

$$y=3\pm6$$

$$\begin{array}{ccc} y=3+6&\qquad\text{or}\qquad &y=3-6\\~\\ y=9&\text{or}&y=-3 \end{array}$$

The possible y-coordinates of the point Alice at are  9  and  -3 .

The sum of the possible y-coordinates of the point Alice is at   =   9 + -3   =   6

To help check the answer, we can see on this graph that  (4, 9)  and  (4, -3) , the two possible points Alice is at, are the same distance away from  (2, 3) ...over 2 units and up or down  6  units.

Aug 19, 2018

#1
+7598
+1

Alice is one the line  x = 4  ,  so the  x-coordinate of whatever point Alice is on is  4 .

Let's call the point that Alice is located  (4, y)  .

By the distance forumula...

the distance between  (4, y)  and  (2, 3)   =   $$\sqrt{(4-2)^2+(y-3)^2}$$

The problem tells us that the distance between  (4, y)  and  (2, 3)  is  √40 ,  so...

$$\sqrt{(4-2)^2+(y-3)^2}=\sqrt{40}$$

Now let's solve this equation for  y .  First square both sides.

$$(4-2)^2+(y-3)^2=40$$

Simplify  $$(4-2)^2$$  to  $$4$$ .

$$4+(y-3)^2=40$$

Subtract  4  from both sides of the equation.

$$(y-3)^2=36$$

Take the ± square root of both sides.

$$y-3=\pm\sqrt{36}$$

$$y-3=\pm6$$

$$y=3\pm6$$

$$\begin{array}{ccc} y=3+6&\qquad\text{or}\qquad &y=3-6\\~\\ y=9&\text{or}&y=-3 \end{array}$$

The possible y-coordinates of the point Alice at are  9  and  -3 .

The sum of the possible y-coordinates of the point Alice is at   =   9 + -3   =   6

To help check the answer, we can see on this graph that  (4, 9)  and  (4, -3) , the two possible points Alice is at, are the same distance away from  (2, 3) ...over 2 units and up or down  6  units.

hectictar Aug 19, 2018