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A triangle with vertices (2, 9), (–5, 1), and (1, –4) is dilated by a scale factor of 4. Which matrix expression represents the transformation?

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Hagrid
This is the correct answer for this question:

First, let's determine the distances of the original triangle's lengths. Let P=(1,4), Q=(2,9), and R=(5,1) be the vertices of the original triangle. 

||PQ||^2 = (2-1)^2 + (9-4)^2 = 1 + 25 = 26 -----> ||PQ|| = sqrt(26) 
||QR||^2 = (5-2)^2 + (1-9)^2 = 9 + 64 = 73 -----> ||QR|| = sqrt(73) 
||RP||^2 = (1-5)^2 + (4-1)^2 = 16 + 9 = 25 -----> ||RP|| = 5 

Now let's do the matrix operation. 

[1 4] [4 0] 
[2 9] [0 4] 
[5 1] 

When you multiply it out, the resulting matrix is: 

[4 16] 
[8 36] 
[20 4] 

Answer with explanation:

The vertices of Triangle are A (2,9) , B(-5,1) and C(1,-4).

Now, the triangle is Dilated by a Scale factor of 4 Units.

When the triangle is dilated by a Scale factor of 4 units, the preimage get Enlarged by unit of 4.

So, if the vertices of Triangle are , (a,b), (p,q) and (m,n) and if it is dilated by a factor of k, then Vertices of enlarged triangle, that is coordinates of vertices of new triangle becomes, (k a,k b),(k p, k q),and ( km,k n).

So, Vertices of Triangle ABC, when enlarged by a Scale factor of 4 are

  =A'(8,36), B'(-20,4) and C'(4,-16).

If you represent vertices of triangle (a,b), (p,q) and (m,n) in Matrix form, it is as follows

→Area of triangle

                  [tex]\left[\begin{array}{ccc}a&b&1\\p&q&1\\m&n&1\end{array}\right] \times \frac{1}{2}[/tex]

→Area of triangle A (2,9) , B(-5,1) and C(1,-4) in matrix form

        [tex]\left[\begin{array}{ccc}2&9&1\\-5&1&1\\1&-4&1\end{array}\right]\times \frac{1}{2}[/tex]

→Area of Dilated triangle in matrix form

    [tex]\left[\begin{array}{ccc}8&36&1\\-20&4&1\\4&-16&1\end{array}\right]\times\frac{1}{2}[/tex]

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