Chapter 4, Problem 6PS

a.

To determine

Explain: How $\Delta ABC,\Delta ADE$ and $\Delta AFG$ are similar

Explanation of Solution

Given information:

  

Formula used:

Angle-Angle-Angle property of similarity of triangles.

Since, $\angle A$ is common in all three triangles.

So, $\angle A=\angle A=\angle A$

Since, $\angle ACB,\angle AED,\angle AGF$ are right angle.

So, $\angle ACB=\angle AED=\angle AGF$

Since, $BC\parallel DE\parallel FG$ with $AF$ as transversal line.

  $\angle ABC=\angle ADE=\angle AFG$ ( corresponding angles)

So, By Angle-Angle-Angle property of similarity of triangles,

  $\Delta ABC\sim \Delta ADE\sim \Delta AFG$

b.

To determine

To find: Similarities imply about ratios $\frac{BC}{AB},\frac{DE}{AD}$ and $\frac{FG}{AF}$ .

Answer to Problem 6PS

Similarities imply about ratios $\frac{BC}{AB},\frac{DE}{AD}$ and $\frac{FG}{AF}$ is $\frac{BC}{AB}=\frac{DE}{AD}=\frac{FG}{AF}$ .

Explanation of Solution

Given information:

  

Formula used:

Corresponding sides of similar triangle are proportional.

As we know, Corresponding sides of similar triangle are proportional.

As, $\Delta ABC\sim \Delta ADE$

So,

  $\begin{array}{l}\frac{BC}{DE}=\frac{AB}{AD}\\ \frac{BC}{AB}=\frac{DE}{AD}\mathrm{.....1}\end{array}$

As, $\Delta ADE\sim \Delta AFG$

So,

  $\begin{array}{l}\frac{DE}{FG}=\frac{AD}{AF}\\ \frac{DE}{AD}=\frac{FG}{AF}\mathrm{.....2}\end{array}$

By equation 1 and 2,

We get,

  $\frac{BC}{AB}=\frac{DE}{AD}=\frac{FG}{AF}$

Similarities imply about ratios $\frac{BC}{AB},\frac{DE}{AD}$ and $\frac{FG}{AF}$ is $\frac{BC}{AB}=\frac{DE}{AD}=\frac{FG}{AF}$ .

c.

To determine

To Find: Does the value of $\sin A$ depend on which triangle is it used to calculated?

Answer to Problem 6PS

No, the value of $\sin A$ does not depend on which triangle it is used to calculated.

Explanation of Solution

Given information:

  

The value of any trigonometric ratio depends on angle.

As the angle is constant in all the three triangles. So, the value of $\sin A$ will not change with triangle.

No, the value of $\sin A$ does not depend on which triangle it is used to calculated.

d.

To determine

To Find: Does the conclusion from part(c) applies to other five trigonometric functions.

Answer to Problem 6PS

Yes, the conclusion from part(c) applies to other five trigonometric functions.

Explanation of Solution

Given information:

  

The value of any trigonometric ratio depends on angle.

As the angle is constant in all the three triangles. So, the value of all trigonometric ratios will not change with triangle.

Yes, the conclusion from part(c) applies to other five trigonometric functions.