Question

review problem

1. these triangles are scaled copies of each other.

for each pair of triangles listed, the area of the second triangle is how many times larger than the area of the first?
a. triangle g and triangle f ____________
b. triangle g and triangle b ____________
c. triangle b and triangle f ____________
d. triangle f and triangle h ____________
e. triangle g and triangle h ____________
f. triangle h and triangle b ____________

Explanation:

Step1: Recall area - ratio formula for similar triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side - lengths. Let the ratio of the corresponding side - lengths of two similar triangles be \(k\), then the ratio of their areas \(A_2/A_1=k^{2}\).

Step2: Find the ratio of side - lengths for each pair

a. Triangle G and Triangle F

The ratio of the corresponding side - lengths of Triangle G to Triangle F: For example, if we take the vertical sides, \(\frac{4}{8}=\frac{1}{2}\). Then the ratio of the area of Triangle G to the area of Triangle F is \((\frac{1}{2})^2=\frac{1}{4}\). So the area of Triangle F is 4 times the area of Triangle G.

b. Triangle G and Triangle B

The ratio of the corresponding side - lengths of Triangle G to Triangle B: Taking the vertical sides, \(\frac{4}{2} = 2\). Then the ratio of the area of Triangle G to the area of Triangle B is \(2^{2}=4\).

c. Triangle B and Triangle F

The ratio of the corresponding side - lengths of Triangle B to Triangle F: Taking the vertical sides, \(\frac{2}{8}=\frac{1}{4}\). Then the ratio of the area of Triangle B to the area of Triangle F is \((\frac{1}{4})^2=\frac{1}{16}\). So the area of Triangle F is 16 times the area of Triangle B.

d. Triangle F and Triangle H

The ratio of the corresponding side - lengths of Triangle F to Triangle H: Taking the vertical sides, \(\frac{8}{\frac{8}{3}} = 3\). Then the ratio of the area of Triangle F to the area of Triangle H is \(3^{2}=9\).

e. Triangle G and Triangle H

The ratio of the corresponding side - lengths of Triangle G to Triangle H: Taking the vertical sides, \(\frac{4}{\frac{8}{3}}=\frac{3}{2}\). Then the ratio of the area of Triangle G to the area of Triangle H is \((\frac{3}{2})^2=\frac{9}{4}\). So the area of Triangle G is \(\frac{9}{4}\) times the area of Triangle H.

f. Triangle H and Triangle B

The ratio of the corresponding side - lengths of Triangle H to Triangle B: Taking the vertical sides, \(\frac{\frac{8}{3}}{2}=\frac{4}{3}\). Then the ratio of the area of Triangle H to the area of Triangle B is \((\frac{4}{3})^2=\frac{16}{9}\).

Answer:

a. \(\frac{1}{4}\)
b. 4
c. \(\frac{1}{16}\)
d. 9
e. \(\frac{9}{4}\)
f. \(\frac{16}{9}\)