Thread: Find h and k

1. Find h and k

Trapezoid TEAM has the following vertices T(-2,3),
E(1,0), A(7,6) and M(0,5). Points E, F(h,k) and A are on the same line. If T, E, F and M are the vertices of a square, find h and k.

MY WORK:

Since points E, F and A lie on the same line, this tells me that they have the same slope, which I found to be 1.

This is where I got stuck.

Where do I go from there?

I love this coordinate geometry stuff.

I realized that point F lies perfectly somewhere on line segment EA to form a square with the other two vertices given. Is this correct?

2. Hello, magentarita!

Please check the wording of the problem.
As written, the facts are wrong.

Trapezoid TEAM has vertices: .$\displaystyle T(-2,3),\;E(1,0),\;A(7,6),\;M(0,5)$

Points E, F(h,k) and A are on the same line.
If $\displaystyle T, E, F, M$ are the vertices of a square, find $\displaystyle h$ and $\displaystyle k.$
Code:
            |
|                       o A
|                 *   *
|           *       *
M|     *           *
o               *
* | .           *
T o   |   .       *
* |     .   *
*       o F
| *   *
- - - - + - o - - - - - - - - - -
|   E
|
Quadrilateral $\displaystyle TEFM$ is not a square.

While $\displaystyle TM \perp TE$, the length of $\displaystyle TM$ is not equal to that of $\displaystyle TE.$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If this problem has multiple questions, okay.
. . Otherwise, it is stupidly written.

. . $\displaystyle \boxed{\begin{array}{c}\text{Three vertices of a rectangle are:}\\ M(0,5),\:T(\text{-}2,3),\:E(1,0)\\ \text{Find }F\text{, the fourth vertex.}\end{array}}$

Who needs that trapezoid?

3. [quote=magentarita;279857]Trapezoid TEAM has the following vertices T(-2,3),
E(1,0), A(7,6) and M(0,5). Points E, F(h,k) and A are on the same line. If T, E, F and M are the vertices of a square, find h and k.

Originally Posted by Soroban
Hello, magentarita!

Please check the wording of the problem.
As written, the facts are wrong.

Code:
            |
|                       o A
|                 *   *
|           *       *
M|     *           *
o               *
* | .           *
T o   |   .       *
* |     .   *
*       o F
| *   *
- - - - + - o - - - - - - - - - -
|   E
|
Quadrilateral $\displaystyle TEFM$ is not a square.

While $\displaystyle TM \perp TE$, the length of $\displaystyle TM$ is not equal to that of $\displaystyle TE.$
Hi Magentarita,

Soroban is absolutely right. Perhaps you meant to say rectangle instead of square.

If you're looking to find the coordinates of F, we can procede this way.

The slope of TM = 1. The slope of EA = the slope of EF = 1. TM is parallel to EF.

The slope of TE is -1. Therefore the slope of MF must be -1.

We can now set up a system of equations to solve for F(h, k).

$\displaystyle \text{Slope of MF}=\frac{k-5}{h-0}=-1$

$\displaystyle k-5=-h$

[1] $\displaystyle h=5-k$

$\displaystyle \text{Slope of EF}=\frac{k-0}{h-1}=1$

$\displaystyle h-1=k$

[2] $\displaystyle h=k+1$

Solving [1] and [2],

$\displaystyle k+1=5-k$

$\displaystyle \boxed{k=2}$

$\displaystyle \boxed{h=5-2=3}$

$\displaystyle \boxed{F(h, k)=F(3, 2)}$

4. yes....

Originally Posted by Soroban
Hello, magentarita!

Please check the wording of the problem.
As written, the facts are wrong.

Code:
            |
|                       o A
|                 *   *
|           *       *
M|     *           *
o               *
* | .           *
T o   |   .       *
* |     .   *
*       o F
| *   *
- - - - + - o - - - - - - - - - -
|   E
|
Quadrilateral $\displaystyle TEFM$ is not a square.

While $\displaystyle TM \perp TE$, the length of $\displaystyle TM$ is not equal to that of $\displaystyle TE.$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If this problem has multiple questions, okay.
. . Otherwise, it is stupidly written.

. . $\displaystyle \boxed{\begin{array}{c}\text{Three vertices of a rectangle are:}\\$$\displaystyle M(0,5),\:T(\text{-}2,3),\:E(1,0)\\ \text{Find }F\text{, the fourth vertex.}\end{array}}$

Who needs that trapezoid?
I agree. The math book states that TEAM is a trapezoid but I suspected that TEFM is a rectangle and not a square. I also think the question is useless.

5. then...

[quote=masters;279998]
Originally Posted by magentarita
Trapezoid TEAM has the following vertices T(-2,3),
E(1,0), A(7,6) and M(0,5). Points E, F(h,k) and A are on the same line. If T, E, F and M are the vertices of a square, find h and k.

Hi Magentarita,

Soroban is absolutely right. Perhaps you meant to say rectangle instead of square.

If you're looking to find the coordinates of F, we can procede this way.

The slope of TM = 1. The slope of EA = the slope of EF = 1. TM is parallel to EF.

The slope of TE is -1. Therefore the slope of MF must be -1.

We can now set up a system of equations to solve for F(h, k).

$\displaystyle \text{Slope of MF}=\frac{k-5}{h-0}=-1$

$\displaystyle k-5=-h$

[1] $\displaystyle h=5-k$

$\displaystyle \text{Slope of EF}=\frac{k-0}{h-1}=1$

$\displaystyle h-1=k$

[2] $\displaystyle h=k+1$

Solving [1] and [2],

$\displaystyle k+1=5-k$

$\displaystyle \boxed{k=2}$

$\displaystyle \boxed{h=5-2=3}$

$\displaystyle \boxed{F(h, k)=F(3, 2)}$
So, what we have hidden in this geometry question is a system of linear equations in two variables h and k, right?