Question

If the middle points of the sides of a triangle are ( 1 , 1 ) , ( 2 , 3 ) and ( 4 , 1 ) , find the co-ordinates of the vertices.​

Answer 1

Answer:

A(-1, 3); B(3, -1); C(5, 3)

Step-by-step explanation:

It’s never a bad idea to sketch the triangle first. If we sketch the midpoints, we can get an idea of what the original triangle looks like.

Let’s label the vertices. Starting from the left, we will have the vertices A, B, and C.

We need to find the coordinates of each vertex. So, we will need to find (a, b) for Vertex A; (c, d) for Vertex B; and (e, f) for Vertex C.

The three given points are the midpoints of the triangle. Thus, we can use the midpoint formula:

$\displaystyle M = \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Since we have three vertices, we will have three midpoints.

Notice that (1, 1) is the midpoint between A and B; (4, 1) is the midpoint between B and C; and (2, 3) is the midpoint between A and C. Therefore, we can write the following three statements:

Statement 1: (1, 1):

The midpoint is (1, 1). By the midpoint formula:

$\displaystyle (1, 1) = \left(\frac{a+c}{2}, \frac{b+d}{2}\right)$

(Remember that A is (a, b) and B is (c, d).)

This yields two equations:  

$\displaystyle \frac{a+c}{2}=1\text{ and } \frac{b+d}{2}=1$

Statement 2: (4, 1):

Likewise:

$\displaystyle (4, 1) = \left(\frac{c+e}{2}, \frac{d+f}{2}\right)$

And again, this yields two equations:

$\displaystyle \frac{c+e}{2}=4\text{ and } \frac{d+f}{2}=1$

Statement 3: (2, 3):

Following the previous steps, this will yield the equations:

$\displaystyle \frac{a+e}{2}=2\text{ and } \frac{b+f}{2}=3$

This yields a total of six equations:

$\displaystyle \frac{a+c}{2}=1\text{ and } \frac{b+d}{2}=1$

$\displaystyle \frac{c+e}{2}=4\text{ and } \frac{d+f}{2}=1$

$\displaystyle \frac{a+e}{2}=2\text{ and } \frac{b+f}{2}=3$

We can multiply each equation by two to simplify:

$a+c=2\text{ and } b+d=2\\c+e=8\text{ and } d+f=2\\a+e=4\text{ and } b+f=6$

The first column are the x-coordinates. The second column are the y-coordinates.

Since none of the variables in the first column (the x-coordinates) repeat in the second column (the y-coordinates) and vice versa, we can solve each column individually. Therefore:

First Column: The X-Coordinates:

We essentially have the triple system of equations:

$\begin{cases}a+c=2, \\c+e=8\\a+e=4\\\end{cases}$

We can solve this using elimination. Multiplying the first equation by negative one yields:

$-a-c=-2$

Adding it to the second:

$(c+e)+(-a-c)=(8)+(-2)$

Simplify:

$-a+e=6$

We can add the newly acquired equation to the third equation:

$(a+e)+(-a+e)=(4)+(6)$

Add:

$2e=10$

Divide. Hence, the value of e is:

$e=5$

This allows us to find the two other values. Using the second equation:

$\displaystyle \begin{aligned} c+e&=8 \\ c+(5)&=8 \\c&=3\end{aligned}$

And the third:

$\begin{aligned} a + e &= 4 \\ a+(5) &= 4\\ a & = -1\end{aligned}$

Therefore, our x-coordinates are:

$a=-1, \; c=3\; \text{ and } e=5$

Second Column: The Y-Coordinates:

We have another system of equations.

$\begin{cases}b+d=2\\d+f=2\\b+f=6\end{cases}$

This can be solved very similarly to the last one. By multiplying the first equation by negative one:

$-b-d=-2$

And adding it to the second:

$\displaystyle \begin{aligned}(d+f)+(-b-d) &= (2)+(-2) \\ f - b &= 0 \end{aligned}$

And adding this to the third:

$\displaystyle \begin{aligned} (b+f) + (f - b) &= (6) + (0) \\ 2f & = 6\end{aligned}$

Hence:

$f=3$

Find the values of d using the second equation:

$\begin{aligned}d+f&=2\\d+(3)&=2\\d&=-1\end{aligned}$

And b using the third:

$\begin{aligned}b+f&=6\\b+(3)&=6\\b&=3\end{aligned}$

Therefore, our y-coordinates are:

$b=3,\; d=-1\; \text{ and } f=3$

And our x-coordinates were:

$a=-1, \; c=3\; \text{ and } e=5$

The vertices of our triangle were given by: A(a, b); B(c, d); and C(e, f).

In conclusion, our vertices are:

A(-1, 3)

B(3, -1)

C(5, 3).

Answer 2

Answer:

A(-1, 3); B(3, -1); C(5, 3)

Step-by-step explanation: