Question

problem 3
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 the area - ratio formula for similar triangles

If the ratio of the corresponding side - lengths of two similar triangles is \(k\), the ratio of their areas is \(k^{2}\).

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

a. For Triangle \(G\) and Triangle \(F\)

The side - length of Triangle \(G\) corresponding to the side of length 6 in Triangle \(F\) is 3. The ratio of the side - lengths \(k_{1}=\frac{3}{6}=\frac{1}{2}\). The ratio of the areas of Triangle \(G\) to Triangle \(F\) is \(k_{1}^{2}=(\frac{1}{2})^{2}=\frac{1}{4}\), so the area of Triangle \(F\) is 4 times larger than the area of Triangle \(G\).

b. For Triangle \(G\) and Triangle \(B\)

The side - length of Triangle \(G\) corresponding to the side of length \(\frac{3}{2}\) in Triangle \(B\) is 3. The ratio of the side - lengths \(k_{2}=\frac{3}{\frac{3}{2}} = 2\). The ratio of the areas of Triangle \(G\) to Triangle \(B\) is \(k_{2}^{2}=4\), so the area of Triangle \(G\) is 4 times larger than the area of Triangle \(B\).

c. For Triangle \(B\) and Triangle \(F\)

The side - length of Triangle \(B\) corresponding to the side of length 6 in Triangle \(F\) is \(\frac{3}{2}\). The ratio of the side - lengths \(k_{3}=\frac{\frac{3}{2}}{6}=\frac{1}{4}\). The ratio of the areas of Triangle \(B\) to Triangle \(F\) is \(k_{3}^{2}=\frac{1}{16}\), so the area of Triangle \(F\) is 16 times larger than the area of Triangle \(B\).

d. For Triangle \(F\) and Triangle \(H\)

The side - length of Triangle \(H\) corresponding to the side of length 6 in Triangle \(F\) is 2. The ratio of the side - lengths \(k_{4}=\frac{2}{6}=\frac{1}{3}\). The ratio of the areas of Triangle \(H\) to Triangle \(F\) is \(k_{4}^{2}=\frac{1}{9}\), so the area of Triangle \(F\) is 9 times larger than the area of Triangle \(H\).

e. For Triangle \(G\) and Triangle \(H\)

The side - length of Triangle \(H\) corresponding to the side of length 3 in Triangle \(G\) is 2. The ratio of the side - lengths \(k_{5}=\frac{2}{3}\). The ratio of the areas of Triangle \(H\) to Triangle \(G\) is \(k_{5}^{2}=\frac{4}{9}\), so the area of Triangle \(G\) is \(\frac{9}{4} = 2.25\) times larger than the area of Triangle \(H\).

f. For Triangle \(H\) and Triangle \(B\)

The side - length of Triangle \(H\) corresponding to the side of length \(\frac{3}{2}\) in Triangle \(B\) is 2. The ratio of the side - lengths \(k_{6}=\frac{2}{\frac{3}{2}}=\frac{4}{3}\). The ratio of the areas of Triangle \(H\) to Triangle \(B\) is \(k_{6}^{2}=\frac{16}{9}\approx1.78\), so the area of Triangle \(H\) is \(\frac{16}{9}\) times larger than the area of Triangle \(B\).

Answer:

a. The area of Triangle \(F\) is 4 times larger than the area of Triangle \(G\).
b. The area of Triangle \(G\) is 4 times larger than the area of Triangle \(B\).
c. The area of Triangle \(F\) is 16 times larger than the area of Triangle \(B\).
d. The area of Triangle \(F\) is 9 times larger than the area of Triangle \(H\).
e. The area of Triangle \(G\) is 2.25 times larger than the area of Triangle \(H\).
f. The area of Triangle \(H\) is \(\frac{16}{9}\) times larger than the area of Triangle \(B\).