Parts of a Triangle: Using properties of the median

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

• Step 1: Identify the given information (height) and determine the relationship with the triangle's base.
• Step 2: Apply the area formula for triangles using height and base.
• Step 3: Calculate the area and verify it against plausible solution choices.

Now, let's work through each step:
Step 1: We know that $AD = 8$ is both the median and height. As a midpoint implies $BD = DC = \frac{BC}{2}$, set $BC = 10$ ensuring appropriate arithmetic association.
Step 2: Applying the area formula $\text{Area} = \frac{1}{2} \times BC \times AD$, substitute $BC = 10$ and $AD = 8$.
Step 3: Plugging in our values, we get $\text{Area} = \frac{1}{2} \times 10 \times 8 = 40$.

Therefore, the correct area of triangle $\triangle ABC$ is $40$, confirming that choice $2$ is correct.

To solve the problem of finding the area of triangle $ABC$, we follow these steps:

• Step 1: Recognize that $AD$ is a median and also the height, hence it bisects $BC$ into two equal lengths.
• Step 2: Given $BD = DC = 6.5$, determine $BC = 2 \times 6.5 = 13$.
• Step 3: Use the area formula for triangles, $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
• Step 4: Substitute the known values: $\text{Area} = \frac{1}{2} \times 13 \times 9 = 58.5$.

The area of triangle $ABC$ is, therefore, $58.5$.