Right Triangle Side Length: Finding BC When Area = 27 and Height = 9

ABC right triangle with an area of 27.

How long is side BC?

To solve this problem, we'll follow these steps:

• Step 1: Use the given information to set up the equation for the area of the triangle.

• Step 2: Calculate the length of side $BC$ using the area formula.

• Step 3: Verify the solution with the given choices.

Now, let's work through each step:
Step 1: We know the area of the right triangle $\triangle ABC$ is given as $27$. The formula for the area of a right triangle is: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
Given that $AB = 9$ can be considered as the base, let $BC$ be the height. Thus, the area formula translates to: $27 = \frac{1}{2} \times 9 \times BC$

Step 2: We solve for $BC$ by rearranging the formula:
$27 = \frac{1}{2} \times 9 \times BC$
$27 = 4.5 \times BC$
$BC = \frac{27}{4.5}$
$BC = 6$

Step 3: According to the calculation, the length of $BC$ is $6$. Reviewing the choices given, the correct answer is option 1: $6$.

Therefore, the length of side $BC$ is $6$.