MATH SOLVE

4 months ago

Q:
# Rectangle 1 has length x and width y. Rectangle 2 is made by multiplying each dimension of Rectangle 1 by afactor of k, where k>0.(a) Are Rectangle 1 and Rectangle 2 similar? Why or why not?(b) Write a paragraph proof to show that the perimeter of Rectangle 2 is k times the perimeter of Rectangle 1.(c) Write a paragraph proof to show that the area of Rectangle 2 is k2 times the area of Rectangle 1.

Accepted Solution

A:

Answer:a) yes they are similarb)Proved belowc)Proved belowStep-by-step explanation:a) rectangle 1 and 2 are similar by definition of similar rectangles.Definition of similar rectangles states that two rectangles are similar if they have same shape but different sizes. The ratio between their corresponding sides are same, in this case as rectangle 2 is made by multiplying each dimension of Rectangle 1 by a
factor of k, where k>0 thus providing same proportion in length of different corresponding sides of two rectangles hence rectangle 1 and 2 are similar.b) Perimeter of rectangle 1, P1= s1+ s2+ s3 + s4As each side of rectangle 2 is formed by multiplying each dimension of Rectangle 1 by k, therefore Perimeter of rectangle 2, P2= k.s1+ k.s2+ k.s3 + k.s4Taking k common = k(s1+ s2+ s3 + s4)Substituting P1 in above expression P2 =k(P1)Hence the perimeter of Rectangle 2 is k times the perimeter of Rectangle 1.c) Area of rectangle 1, A1= 2(s1)(s2)As each side of rectangle 2 is formed by multiplying each dimension of Rectangle 1 by k, therefore Area of rectangle 2, A2= 2(k.s1)(k.s2) = 2k^2(s1)(s2) = k^2 [2(s1)(s2)]Substituting A1 in above expression A2 = k^2(A1)Hence the area of Rectangle 2 is k2 times the area of Rectangle 1 !