Question

PLEASE HEEELLLPP !! IMPORTANT

Answer 1

1. Contractor.

1. a) Did the contractor increase the area by the amount Fernando wanted? Explain.

In this problem, Fernando wants to increase his dog's play yard. This original yard has a fence represented by rectangle LMNO. Fernando wants the contractor to construct a new fence that should enclose 6 times as much area as the current fence. In this way, we must calculate the area of both shapes to compare and conclude whether or not they meet the requirement. So:

FOR THE ORIGINAL FENCE:

To find the lengths of the rectangle, we must count the number of square from O to L (L1) and from O to N (L2):

$A=L_{1}\times L_{2} \\ \\ L_{1}=2 \\ \\ L_{2}=3 \\ \\ A=2(3)=6units^2$

FOR THE EXPANDED FENCE:

To find the lengths of this shape, we must count the number of square from O' to L' (L1) and from O' to N' (L2):

$A=L_{1}\times L_{2} \\ \\ L_{1}=6 \\ \\ L_{2}=6 \\ \\ A'=6(6)=36units^2$

Since A' is 6 times as much area as the current fence, that is, A' = 6A, then the contractor increases the area by the amount Fernando wanted.

1. b) Does the new shape maintain the shape of the old fence? Explain.

Definitely not! Why? because the original fence is a rectangle while the new fence is a square. Notice that for the original fence we have side lengths:

$L_{1}=2 \\ \\ L_{2}=3$

And for the new fence:

$L_{1}=6 \\ \\ L_{2}=6$

So, a rectangle is a quadrilateral whose angles are all right angles while a square is a rectangle whose sides all have the same length. To maintain the shape, both fences have to be proportional.

2. Dilation.

Dilation is when you stretch something by the same amount in two perpendicular directions. In other words, dilation changes size, not overall shape. Dilation factors greater than one makes the shape greater while dilation factors less than one makes the shape smaller. In this exercise, the scale factor equals 1.5, which is greater than one. So the resultant triangle will be greater than the original. Since the center of dilation is the origin, we just need to multiply each coordinate of the triangle by the scale factor. Therefore:

A(0,0) → A'(0 x 1.5, 0 x 1.5) = A'(0,0)

B(5,0) → B'(5 x 1.5, 0 x 1.5) = B'(7.5, 0)

C(5,4) → C'(5 x 1.5, 4 x 1.5) = C'(7.5, 6)

The original coordinates are A, B, C.

The dilated coordinates are A', B', C'