Chapter 8.4, Problem 61PPS

(a)

To determine

To sketch: Three triangles that is not similar to each other and labeled them $ABC,MNP\&XYZ$

Explanation of Solution

Given information: The name of the triangles is given that is $ABC,MNP\&XYZ$ .

Sketch: The required triangles are drawn below.

  

(b)

To determine

To fill: The table.

Explanation of Solution

Given information: The following table is given,

  

The following table is been filled

  

(c)

To determine

To explain: The sum of the squares of the cosine and sine of an acute angle of a right triangle.

Answer to Problem 61PPS

The answer is $1$ .

Explanation of Solution

Given information: The given triangle is a right triangle.

The sum of the squares of cosine and sine of an acute angle is $1$ .

(d)

To determine

To explain: Conjecture about angle $x$ .

Answer to Problem 61PPS

  ${\sin }^{2}x+{\cos }^{2}x=1$

Explanation of Solution

Given information: The given triangle is a right triangle.

The sum of the squares of cosine and sine of an acute angle is $1$ .

The algebraically expression for angle $x$ ,

  ${\sin }^{2}x+{\cos }^{2}x=1$

(e)

To determine

To show: Conjecture about is valid for angle A in the given figure.

Explanation of Solution

Given information: The following figure is been given.

  

From the figure,

  $\begin{array}{l}\sin A=\frac{y}{t}\\ \cos A=\frac{x}{t}\end{array}$

Hence,

  $\begin{array}{l}{\sin }^{2}x+{\cos }^{2}x={\left(\frac{y}{t}\right)}^2+{\left(\frac{x}{t}\right)}^2\\ =\frac{y^2}{t^2}+\frac{x^2}{t^2}\\ =\frac{x^2+y^2}{t^2}\end{array}$

By using Pythagoras Theorem, $x^2+y^2=t^2$

Now, make substitution,

  $\therefore \frac{t^2}{t^2}=1$