Rectangle ABCD

A is (-1,4), B is (4,8)

Equation of BC is 4y + 5x= 52

equation of AD is 4y + 5x = 11

equation of AB is 5y= 4x+ 24

Find C and D

Answer is (12,-2) and (7,-6)

I've tried gradient,length,midpoint formulas but to no avail.

Thanks

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- Aug 13th 2013, 02:23 AMRodimus79Finding 2 vertices of rectangle with 2 known vertices
Rectangle ABCD

A is (-1,4), B is (4,8)

Equation of BC is 4y + 5x= 52

equation of AD is 4y + 5x = 11

equation of AB is 5y= 4x+ 24

Find C and D

Answer is (12,-2) and (7,-6)

I've tried gradient,length,midpoint formulas but to no avail.

Thanks - Aug 13th 2013, 03:29 AMMarkFLRe: Finding 2 vertices of rectangle with 2 known vertices
The problem, as posed, has no unique solution. Is there some other attribute of the rectangle which you have not posted, such as its area?

- Aug 13th 2013, 05:18 AMRodimus79Re: Finding 2 vertices of rectangle with 2 known vertices
Apologies..I left out

Length of BC is twice the length Of AB - Aug 13th 2013, 06:11 AMHallsofIvyRe: Finding 2 vertices of rectangle with 2 known vertices
And, you now tell us, "Length of BC is twice the length Of AB"

It is easy to calculate that the length of AB is $\displaystyle \sqrt{(4-(-1)^2+ (8- 4)^2}= \sqrt{25+ 16}= \sqrt{41}$

so the length of BC is $\displaystyle 2\sqrt{41}$ and, because this is a rectangle, BC and AD are the same length.

Take D= (x, y)= (x, (11- 5x)/4). Then we must have [tex]\sqrt{(x- (-1))^2+ ((11- 5x)/4- 4)^2}= 2\sqrt{41}[tex]

so that, squaring both sides gives $\displaystyle (x+1)^2+ (5x+ 5)^2/16= 164$. Solve that quadratic equation for x. Do the same with BC.

Quote:

Find C and D

Answer is (12,-2) and (7,-6)

I've tried gradient,length,midpoint formulas but to no avail.

Thanks

- Aug 13th 2013, 09:35 AMRodimus79Re: Finding 2 vertices of rectangle with 2 known vertices
Thanks so much, HallsofIvy!! The key step was to have y = 11-5x/4, to express y in terms of x, since it's a point on the line and satisifies the equation! Thanks once again!!!

- Aug 14th 2013, 07:58 PMibduttRe: Finding 2 vertices of rectangle with 2 known vertices
- Aug 16th 2013, 05:49 AMbjhopperRe: Finding 2 vertices of rectangle with 2 known vertices
given the two points and the fact that the length of the rectangle is twice width(AB) there is a simple method to find the other vertices

slope diagram AB = 4/5

slope diagram AD =-5/4

actual slope diagram AD =-10/8 since AD =2AC

Point D x =8-1 =7 y =4 + absolute value y =10 y=-6

same procedure gives point C - Aug 17th 2013, 01:41 AMibduttRe: Finding 2 vertices of rectangle with 2 known vertices
- Aug 17th 2013, 03:45 AMbjhopperRe: Finding 2 vertices of rectangle with 2 known vertices