A=77°
B=24°
C=16°
D=77°
E=40°
F=63°
∆AB=22
∆BC=24
∆CA=16
∆DE=11
∆EF=12
∆FD=8
∆ABC is similar to ∆DEF. Find the ratio between ∆DEF and ∆ABC. Express the answer as a fraction in simplest form
To determine the ratio between ∆DEF and ∆ABC, we need to find the corresponding side lengths of the two triangles.
First, we can use the Law of Sines to find the side lengths of ∆ABC:
AB/sin(A) = BC/sin(B) = CA/sin(C)
22/sin(77°) = 24/sin(24°) = 16/sin(16°)
Using a calculator, we find that:
AB ≈ 22.047
BC ≈ 15.723
CA ≈ 6.793
Similarly, we can use the Law of Sines to find the side lengths of ∆DEF:
DE/sin(D) = EF/sin(E) = FD/sin(F)
11/sin(77°) = 12/sin(40°) = 8/sin(63°)
Using a calculator, we find that:
DE ≈ 10.316
EF ≈ 8.618
FD ≈ 7.733
Now, we can calculate the ratio between the two triangles as the ratio of corresponding side lengths:
∆DEF / ∆ABC = (DE/AB) * (EF/BC) * (FD/CA)
= (10.316/22.04) * (8.618/15.723) * (7.733/6.793)
Calculating these values, we get:
∆DEF / ∆ABC ≈ 0.468
The ratio between ∆DEF and ∆ABC is approximately 0.468.