Chapter 7.4, Problem 4CE

To determine

Statement: Whether the two triangle ABC and DEF are equal and $\text{AB:DE = EF:BC}$ .

Answer to Problem 4CE

  $\Delta \text{ABC}\sim \Delta \text{DEF}$and $\frac{\text{AB}}{\text{DE}}\ne \frac{\text{EF}}{\text{BC}}$ .

Explanation of Solution

Given information: Two triangles are given and $\Delta \text{ABC}\sim \Delta \text{DEF}$ .

  

According to the theory of similar triangles, the triangles may have the same shape, but it is not necessary that they are of same size. Their corresponding angles must be equal and sides must be in the same ratio.

Now, since $\Delta \text{ABC}\sim \Delta \text{DEF}$ , we can write −

  $\frac{\text{AC}}{\text{DF}}=\frac{\text{AB}}{\text{DE}}=\frac{\text{BC}}{\text{EF}}$ , and $\angle \text{M = }\angle {\text{N = 90}}^{·}$ .

Also, $\angle \text{A = }\angle \text{D}$ .

Therefore,

  $\Delta \text{ABC}\sim \Delta \text{DEF}$ , $\frac{\text{AC}}{\text{DF}}=\frac{\text{AB}}{\text{DE}}=\frac{\text{BC}}{\text{EF}}$ , and $\frac{\text{AB}}{\text{DE}}=\frac{\text{EF}}{\text{BC}}$ is not true.