Chapter 15.2, Problem 14PSB

(a)

To determine

To Find: $\text{The side is of an acute, obtuse, right angle triangle or no triangle}\text{.}$

Answer to Problem 14PSB

  $\text{Acute triangle}\text{.}$

Explanation of Solution

Given:

  $\text{ sides of }\Delta 12,13,14$

Formula Used:

Sum of two sides should be always greater than the third side

Pythagorean Theorem:

  $\begin{array}{l}\text{ for acute }\Delta \\ a^2+b^2>c^2\end{array}$

Calculation:

  $\begin{array}{l}{a}^2+b^2>c^2\\ {12}^2+{13}^2>{14}^2\\ 313>196\\ \text{hence the }\Delta \text{ is an acute triangle}\text{.}\end{array}$

Conclusion:

These sides are of an acute triangle.

(b)

To determine

To Find: $\text{The side is of an acute, obtuse, right angle triangle or no triangle}\text{.}$

Answer to Problem 14PSB

  $$

  $\text{Obtuse triangle}$

Explanation of Solution

Given:

  $\text{sides of }\Delta \text{ 11,5,18}$

Formula Used:

Pythagorean Theorem:

  $\begin{array}{l}\text{for obtuse triangle}\\ a^2+b^2<c^2\end{array}$

Calculation:

  $\begin{array}{l}{a}^2+b^2<c^2\\ 5^2+{11}^2<{18}^2\\ 146<324\\ \text{hence the }\Delta \text{ is an obtuse triangle}\text{.}\end{array}$

Conclusion:

Thus the given sides are of the obtuse angle triangle.

(c)

To determine

To Find: $\text{The side is of an acute, obtuse, right angle triangle or no triangle}\text{.}$

Answer to Problem 14PSB

  $\begin{array}{l}\text{Obtuse Triangle}\end{array}$

Explanation of Solution

Given:

  $\text{sides of }\Delta \text{ 15,9,18}$

Formula Used:

Pythagorean Theorem:

  $\begin{array}{l}\text{for obtuse triangle}\\ a^2+b^2<c^2\end{array}$

Calculation:

  $\begin{array}{l}{a}^2+b^2<c^2\\ {15}^2+9^2<{18}^2\\ 306<324\\ \text{hence the }\Delta \text{ is an obtuse triangle}\text{.}\end{array}$

Conclusion:

Thus the given sides are of an obtuse triangle.

(d)

To determine

To Find: $\text{The side is of an acute, obtuse, right angle triangle or no triangle}\text{.}$

Answer to Problem 14PSB

  $\text{Right Angle Triangle}$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Sides of the }\Delta \\ \frac{1}{2},1\frac{1}{5},1\frac{3}{10}\end{array}$

Formula Used:

Pythagorean Theorem:

  $\begin{array}{l}\text{For right angle triangles}\\ {\text{a}}^2+b^2=c^2\end{array}$

Calculation:

  $\begin{array}{l}{\text{a}}^2+b^2=c^2\\ {\left(\frac{1}{2}\right)}^2+{\left(1\frac{1}{5}\right)}^2={\left(1\frac{3}{10}\right)}^2\\ \frac{1}{4}+\frac{36}{25}=\frac{169}{100}\\ \text{⇒}\frac{169}{100}=\frac{169}{100}\\ \text{hence the }\Delta \text{ is a right angle traingle}\text{.}\end{array}$

Conclusion:

Thus the given sides are of the right angled triangle.