Chapter 7.4, Problem 25WE

To determine

To Prove: that $\frac{AD}{XW}=\frac{AB}{XY}$ in the figure shown such that $\Delta ABC\sim \Delta XYZ,\overrightarrow{AD}\text{ and }\overrightarrow{AB}\text{ are altitudes}$ .

Explanation of Solution

Given:

Figure is shown as.

  

  $\Delta ABC\sim \Delta XYZ,\overrightarrow{AD}\text{ and }\overrightarrow{AB}\text{ are altitudes}$

Calculation:

Since, $\Delta ABC\sim \Delta XYZ$ therefore congruent parts of congruent triangle are congruent.

I.e.

  $\angle B\cong \angle Y$

In both triangles ABD and XYW.

  $\begin{array}{l}\angle ADB=\angle XWY\therefore \left(\text{Right angles}\right)\\ \angle B=\angle Y\therefore \left(\text{Proved above}\right)\end{array}$

Therefore, by AA postulate $\Delta ABD\sim \Delta XYW.$

Ratio of sides of two similar triangle is equal.

  $\frac{AD}{XW}=\frac{AB}{XY}$

So, it has been proved.