Chapter 7.5, Problem 18WE

To determine

To prove: The length of corresponding diagonals are in the same ratio to the length of corresponding sides.

Explanation of Solution

Given information:

Below quadrilaterals are similar.

  

Consider two similar quadrilaterals as shown below.

  

Since the above two quadrilaterals are similar, it implies that the corresponding sides are in the same ratio and also the corresponding angles are congruent.

  $\frac{AB}{EF}=\frac{BC}{FG}=\frac{CD}{GH}=\frac{DA}{HE}\dots \dots (1)$

Also, it follows that,

  $\angle A=\angle E,\angle B=\angle F,\angle C=\angle G,\angle D=\angle H$ .......(2)

It is required to prove that the lengths of the corresponding diagonals are in the same ratio as the length of the corresponding sides.

In view of (1), it is sufficient to show that,

  $\frac{BD}{FH}=\frac{BC}{FG}$

Note that in $\text{Δ}CBD$ and $\text{Δ}GFH$

  $\frac{BC}{FG}=\frac{CD}{GH}($ From(1))

  $\angle C=\angle G$ (From(2))

This implies that $\text{Δ}CBD\sim \text{Δ}GFH$

Since the corresponding sides in a similar triangles are proportional.

This implies that,

  $\frac{BD}{FH}=\frac{BC}{FG}=\frac{CD}{GH}$

That is, The length of corresponding diagonals are in the same ratio to the length of corresponding sides.

The ratio of other two diagonals $\frac{AC}{EG}$ can be similarly obtained.