Chapter 6.1, Problem 56E

(a)

To determine

Tofind:The value of $a$ for which one triangle be formed.

Explanation of Solution

There are four cases possible.

Case I:

The vertical distance from point $\left(4,3\right)$ to the positive $x$ axis is $3$ . Thus if $a<3$ then there will be no triangle.

Case II:

If $a=3$ , then the there will be one angle triangle, that will be the right angled triangle.

Case III:

If $3<a<5$ , the a person can either swing the line segment to the right or to the left to intersect the positive $x$ axis. Thus there shall be two triangles.

Case IV :

If $a\ge 5$ , then the person can swing segment a to the right, but not to the left because swinging to the left shall be to long and it will either coincide with the black segment or intersect with negative $x$ axis. But only positive intersection is required.

So, in this case only one triangle will be formed.

One triangle will be formed when,

  $a=3\text{or}a\ge 5$

Therefore, the value of $a$ for which only one triangle is formed are $a=3\text{or}a\ge 5$ .

(b)

To determine

To find:The value of $a$ for which two triangles be formed.

Explanation of Solution

There are four cases possible.

Case I:

The vertical distance from point $\left(4,3\right)$ to the positive $x$ axis is $3$ . Thus if $a<3$ then there will be no triangle.

Case II:

If $a=3$ , then the there will be one angle triangle, that will be the right angled triangle.

Case III:

If $3<a<5$ , the a person can either swing the line segment to the right or to the left to intersect the positive $x$ axis. Thus there shall be two triangles.

Case IV :

If $a\ge 5$ , then the person can swing segment a to the right, but not to the left because swinging to the left shall be to long and it will either coincide with the black segment or intersect with negative $x$ axis. But only positive intersection is required.

So, in this case only one triangle will be formed.

Two triangle will be formed when,

  $3<a<5$

Therefore, the value of $a$ for which only two triangles are formed are $3<a<5$ .

(c)

To determine

To find:The value of $a$ for which no triangles be formed.

Explanation of Solution

There are four cases possible.

Case I:

The vertical distance from point $\left(4,3\right)$ to the positive $x$ axis is $3$ . Thus if $a<3$ then there will be no triangle.

Case II:

If $a=3$ , then the there will be one angle triangle, that will be the right angled triangle.

Case III:

If $3<a<5$ , the a person can either swing the line segment to the right or to the left to intersect the positive $x$ axis. Thus there shall be two triangles.

Case IV :

If $a\ge 5$ , then the person can swing segment a to the right, but not to the left because swinging to the left shall be to long and it will either coincide with the black segment or intersect with negative $x$ axis. But only positive intersection is required.

So, in this case only one triangle will be formed.

No triangles will be formed when,

  $a<3\text{or}0<a<3$

Therefore, the value of $a$ for which only no triangles are formed are $a<3\text{or}0<a<3$ .