Chapter 6.1, Problem 62E

To determine

The values of the a to form the one triangle, two triangles and no triangle.

Explanation of Solution

Given:

The triangle is formed by drawing the line segment of the length a from the (4,3) to the positive axis:

  

The value of the a needs to be evaluated to obtain the different condition:

Referring to the given diagram, the angle B of the triangle ABC is obtuse angle:

Since, the AC=b:

  $\begin{array}{l}b=\sqrt{{\left(4-0\right)}^2+{\left(3-0\right)}^2}\\ =\sqrt{16+9}\\ =5\end{array}$

The triangle will have the one solution when it satisfy the condition a>b.

Here, this triangle will have the one solution if it will satisfy the condition a>5.

Again this triangle have no solution when it satisfies the condition $a\le b$ .

Then this triangle will have no solution when $a\le 5$ .

The required value of the >5 and $a\le 5$ respectively.