A rectangle has three of its vertices at points #(3,4), # #(5,–4),# and #(–7, –7).# What is the #y# coordinate of the unknown vertex?

Question

A rectangle has three of its vertices at points #(3,4), # #(5,–4),# and #(–7, –7).#  What is the #y# coordinate of the unknown vertex?

Maverick Smith · Accepted Answer

The #y#-coordinate of the fourth vertex is #1#.

The given points can be drawn in the #xy#-plane as follows: graph{((x-3)^2+(y-4)^2-0.05)((x-5)^2+(y+4)^2-0.05)((x+7)^2+(y+7)^2-0.05)=0 [-14.95, 13.52, -8.37, 5.87]} From here, it is easy to visualize roughly where the fourth vertex will be—somewhere in #"Quadrant II"#. Since rectangles are composed of two sets of parallel sides, the #y# distance between two adjacent points will be the same as the #y# distance between the other two points. What does this mean? Well, the point #(3,4)# is 8 up from its neighbour #(5,"-"4)# (since #4-"(-4)"=8#), so the fourth vertex will also be 8 up from #("-7","-7")#. And what is 8 up from -7? That's right: 1. We can use this to find the #x#-coordinate as well; just use the #x#-coordinates instead of the #y#'s. The remaining vertex is at #("-9",1)#. Hope this helps!

Sadie Adams · Answer

#(-9, 1#)

Use : The vector #(x_1, y_1) to (x_2, y_2)# is # < x_2-x_1, y_2-y_1>#. The vertices are labelled #A( 3, 4 ), B( 5, -4 ), C(-7. -7) and D( x, y ).#. Vector #AD = < x-3, y-4># = vector # BC=<-12, -3>#. Solving, (#x, y) = (-9, 1)#.

Ella Armstrong · Answer

undefined To find the y-coordinate of the unknown vertex of the rectangle, we first need to determine the length of one of its sides. We can do this by calculating the distance between two known vertices. Let's choose the vertices (3,4) and (5,-4).

Using the distance formula, the distance between these two points is:

$$ \sqrt{((5-3)^2 + (-4-4)^2)} = \sqrt{4^2 + 8^2} = \sqrt{16 + 64} = \sqrt{80} $$

Since a rectangle has equal opposite sides, this distance represents the length of one side of the rectangle.

Now, let's find the vector between the two known vertices: (3,4) and (5,-4). This vector will give us the direction from one point to the other.

$$ \text{Vector} = (5-3, -4-4) = (2, -8) $$

To find the unknown vertex, we add this vector to one of the known vertices. Let's add it to the vertex (3,4):

$$ \text{Unknown Vertex} = (3,4) + (2, -8) = (3+2, 4-8) = (5, -4) $$

So, the y-coordinate of the unknown vertex is -4.