Chapter 10.1, Problem 21E

To determine

To find the side lengths of the given triangle and classify the triangle. To analyze if it is a right triangle.

Answer to Problem 21E

1. The side length measures are $49,168$

The triangle is a scalene triangle.

2. The triangle is a right-angled triangle

Explanation of Solution

Given information:

Ratio of the side length measures in a triangle is given as $7:24:25$

Perimeter of the triangle is given as 392 inches

Calculation:

Let the side length measurements be 7x, 24x, 25x.

The perimeter of the triangle is given as 392 inches. Hence,

  $7x+24x+25x=392$

Adding all the xtogether,

  $56x=392$

Divide both the sides by 56

  $\frac{56x}{56}=\frac{392}{56}$

This will give the value of $x=7$

Hence the length of the side measures are

First side is 7x ,multiply 7 with 7

  $x=7\times 7$ gives $x=49$ inches

Second sideis 24x ,multiply 24 with 7

  $x=24\times 7$ gives $x=168$

Third side is 25x ,multiply 25 with 7

  $x=25\times 7$ gives $x=175$

As can be seen all the length of the side measures are different. Hence the triangle is Scalene triangle.

To check, whether or not the triangle is a right-angled triangle, ,make use of Pythagoras Theorem. According to Pythagoras theorem: $a^2+b^2=c^2$

Let side1 = 49 inches be a

Let side2 = 168 inches be b

Let side3 = 175 inches be c

As can be seen, side3 is the longest side. ,now check if the side measures satisfy the Pythagoras theorem or not.

Side1 : $a^2=49\times 49=2401$

Side2 : $b^2=168\times 168=28224$

Side3 : $c^2=175\times 175=30625$

Hence, according to the Pythagoras theorem: $a^2+b^2=c^2$ ,

  $2401+28224=30625$

As can be seen, $a^2+b^2=c^2$ hence the three side measures satisfy the Pythagoras Theorem and the triangle can be classified as a Right-Angled Triangle

Conclusion- The side measures of the triangle are $49,168,175$ inchesand the triangle can be classified as Scalenetriangle.

Since, the three side measures satisfy the Pythagoras Theorem and the triangle can be classified as a Right-Angled Triangle