Chapter 6.5, Problem 10CE

a.

To determine

To draw the diagram and to find the length of the hypotenuse.

Answer to Problem 10CE

The length of the hypotenuse is 10 centimeters.

Explanation of Solution

Given:

Right triangle with legs of $6$ centimeter and $8$ centimeter.

Formula used:

Pythagoras theorem

  ${\text{hypotenuse}}^2={\text{base}}^{\text{2}}+{\text{height}}^{\text{2}}$

Calculation:

We need to draw a right triangle with legs of $6$ centimeter and $8$ centimeter.

So, the diagram will be:

  

Note: cm represents centimeter.

Where, base $=8$ centimeter and height $=6$ centimeter.

Pythagoras theorem

  $\Rightarrow {\text{hypotenuse}}^2={\text{base}}^{\text{2}}+{\text{height}}^{\text{2}}$

Putting in the values,

  $\Rightarrow {\text{hypotenuse}}^2=8^2+6^2$$\left[ \because {\text{hypotenuse}}^2={\text{base}}^{\text{2}}+{\text{height}}^{\text{2}}\right]$

  $\Rightarrow {\text{hypotenuse}}^2=64+36$

  $\Rightarrow {\text{hypotenuse}}^2=100$

  $\Rightarrow \text{hypotenuse}=\sqrt{100}$

  $\Rightarrow \text{hypotenuse}=10$ centimeter

Thus, the new diagram is,

  

Note: cm represents centimeter.

Conclusion:

Therefore, the diagram is discussed above and the hypotenuse length is $10$ centimeter.

b.

To determine

To determine the length of the third side.

Answer to Problem 10CE

The length of the third side is 4.81 centimeters.

Explanation of Solution

Given:

The legs of the triangle are now of lengths $6$ centimeter and $8$ centimeter, but the angle is reduced to $45°$ .

In the previous part the value of the hypotenuse length is $10$ centimeter.

  

From here, let’s consider $\angle C=\theta$

  $\Rightarrow \tan \theta =\frac{\text{height}}{\text{base}}$

  $\Rightarrow \tan \theta =\frac{6}{8}$

  $\Rightarrow \tan \theta =\frac{3}{4}$

  $\Rightarrow \theta ={\tan }^{-1}\left(\frac{3}{4}\right)$

  $\Rightarrow \theta =36.86$

Now reducing the angle to $45°$ ,

  

Using Law of Cosines:

  $\Rightarrow h^2={\text{height}}^{\text{2}}+{\text{base}}^{\text{2}}-2\times \text{height}\times \text{base}\times cos(c)$

  $\Rightarrow h^2=6^2+8^2-2\times 8\times 6cos(36.86)$

  $\Rightarrow h^2=100-96\times cos(36.86)$

  $\Rightarrow h=4.81$ centimeter

Conclusion:

Therefore, the length of the third side $4.81$ centimeter.