Question

Dilate a square with vertices $\left(0,\ 0\right)$(0, 0)​ , $\left(0,\ 4\right)$(0, 4)​ , $\left(4,\ 4\right)$(4, 4)​ , and $\left(4,\ 0\right)$(4, 0)​ using the scale factor $k=0. 5$k=0. 5​. What is the value of the ratio (new to original) of the perimeters? the areas?

Answer 1

Using coordinates rule of Dilation , The ratio (new to original) of perimeter is 1/√2

The ratio of (new to original) of area is ½

The coordinates of the vertices of Square are (0,0),(0,4),(4,0) and (4,4).

Let the vertices be

A(0,0),B(0,4),C(4,0),D(4,4) of square ABCD. Also , scaling factor k= 0.5 or ½ .

Now, we shall find out the coordinates of square KLMN which is image of Square ABCD.

Use the coordinates rule for dilation with k= ½

(x,y) → (kx , ky)

(kx , ky ) represents coordinates of new square KLMN.

(0,0) →(0,0)=K(0,0)

(0,4)→ ( 0×(1/2), 4(1/2))= L(0,2)

(4,0)→( 4(1/2), 0(1/2)) = M(2,0)

(4,4)→( 4(1/2), 4(1/2)) =N (2,2)

Side of square ABCD is find out by distance formula i.e., d= √square of difference of x-coordinates + square of difference of y-coordinates

Length of AB = √ 0+4 = 2unit

Perimeter of Square of ABCD = 8 units

area of Square ABCD = 4 square units

Length of KL = √0+2 = √2 units

Perimeter of Square KLMN= 4√2 units

Area of Square KLMN= 2 square units

Ratio of ( new to original) of perimeter = 4√2/8 = 1/√2

Ratio of ( new to original) of area = 2/4 = 1/2

To learn more about dilation , refer :

brainly.com/question/3123885

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