Questions: Round to the nearest thousandth if necessary. 16. a =? b = 12 km c = 15 km

Transcript text: Round to the nearest thousandth if necessary. 16. $\begin{aligned} a & =? \\ b & =12 \mathrm{~km} \\ c & =15 \mathrm{~km}\end{aligned}$

Solution

Solution Steps

To find the value of $ a $ in a right triangle where $ b $ and $ c $ are given, we can use the Pythagorean theorem. The Pythagorean theorem states that $ a^2 + b^2 = c^2 $. We can solve for $ a $ by rearranging the equation to $ a = \sqrt{c^2 - b^2} $. Finally, we will round the result to the nearest thousandth.

Step 1: Apply the Pythagorean Theorem

Given the right triangle with legs $ b = 12 \, \text{km} $ and hypotenuse $ c = 15 \, \text{km} $, we can use the Pythagorean theorem, which states: \[ a^2 + b^2 = c^2 \] We need to solve for $ a $: \[ a^2 = c^2 - b^2 \]

Step 2: Substitute the Values

Substituting the known values into the equation: \[ a^2 = 15^2 - 12^2 \] Calculating the squares: \[ a^2 = 225 - 144 \]

Step 3: Calculate $ a^2 $

Now, we find: \[ a^2 = 81 \]

Step 4: Solve for $ a $

Taking the square root of both sides gives: \[ a = \sqrt{81} = 9.0 \]