Question

if $\triangle abc \sim \triangle amn$ and $am = 6$, $mb = 4$, $an = 8$, then what is the value of $nc$?
(options: 3, 5.3, 7.3, 7)

Explanation:

Step1: Find AB length

AB is the sum of AM and MB. So \( AB = AM + MB = 6 + 4 = 10 \).

Step2: Use similar triangles ratio

Since \( \triangle ABC \sim \triangle AMN \), the ratios of corresponding sides are equal. So \( \frac{AM}{AB}=\frac{AN}{AC} \).
We know \( AM = 6 \), \( AB = 10 \), \( AN = 8 \). Let \( AC = AN + NC = 8 + NC \). Substitute into the ratio: \( \frac{6}{10}=\frac{8}{8 + NC} \).

Step3: Solve for NC

Cross - multiply: \( 6(8 + NC)=10\times8 \).
Expand: \( 48+6NC = 80 \).
Subtract 48 from both sides: \( 6NC=80 - 48 = 32 \).
Then \( NC=\frac{32}{6}=\frac{16}{3}\approx5.3 \).

Answer:

\( \frac{16}{3}\) (or approximately \( 5.3 \))