Three vertices of a rectangle are (0,4), (2,1), and (8,5).?

Question

Find the coordinates of the 4th vertex. Show work.

Answer 1

(10,2)

I plotted the given points on the plane. From (0,4) to (2,1), you go down 3, right 2. So from (8,5), you go down 3, right 2 and end up at (10,2)

There is another answer, but I will let you find it.

Answer 2

Hi Dear ZZ Z

■ Step 1
- as you know in every rectangle the parallel lines have a same slope ; in our rectangle if the 4th vertex we are looking for is ( p, q) the slope of line (0,4) & (2,1) equall slope of line (p ,q ) & (8,5)
{ slope formula = ( y2 - y1) / ( x2 - x1) } ...

( q - 5 ) / ( p - 8 ) = ( 1 - 4 ) / ( 2 - 0 )
( q - 5 ) / ( p - 8 ) = -3 / 2
( q - 5 )*2 = ( p - 8 )* -3
2q - 10 = -3p+ 24
2q + 3p = 24+ 10
2q + 3p = 34

■ Step 2
the slope of line (0,4) & (8,5) equall slope of line (2 ,1 ) & (p , q)
( 5 - 4 ) / ( 8 - 0 ) = ( q - 1 ) / ( p - 2 )
1 / 8= ( q - 1 ) / ( p - 2 )
( p - 2) * 1 = 8 * ( q - 1)
p -2 = 8q - 8
p - 8q = - 6

■ Step 3
solve these equations
2q + 3p = 34
p - 8q = - 6 { -3 * ( p - 8q = - 6 ) = -3p + 24 q = + 18 }

2q + 3p = 34 - ( 3p + 24 q = + 18 )
26q = 52 , q = +2

if q = 2 , p - 8*2 = - 6 , p = -6 + 16 , p = +10
So p= + 10 & q = +2
( +10 , +2 ) 4th vertex

Good Luck Dear ♣

Answer 3

treat the sides as if they were lines. the slope of the side from (2,1) to (8,5) is (5-1)/(8-2)=4/6=2/3, so the other side must be parallel to it and also have a slope of 2/3. the other side has an endpoint at (0, 4), so the line has a y-intercept of 4. the line thus has an equation of y=2/3x+4. since the two sides are the same length, the other endpoint must be 6 units to the right, so its y-coordinate must be y=2/3(6)+4=4+4=8. therefore, the other vertex is at (6, 8)

Answer 4

Let (x,y) be the 4th vertex. Consider slopes of parallel sides.

Slope of line from (x,y) to (8,5) = slope of line from (2,1) to (0,4)

( y - 5 ) / ( x - 8 ) = ( 1 - 4 ) / ( 2 - 0 )

Simplify to get

3x + 2y = 34

Similarly for the other parallel sides, to get

x - 8y = -6

Solve these equations to get

x = 10, y = 2

Answer 5

the 3rd ver is 2