Parts of a Triangle: Using properties of the median

Given that $AD$ is a median in triangle $\triangle ABC$ and $BD = 6$, we need to determine the length of side $BC$.

Since $AD$ is the median, it implies that $D$ is the midpoint of $BC$. By definition of midpoint, this means:

• $BD = DC$.
• Given $BD = 6$, it follows that $DC = 6$ as well, since $D$ divides $BC$ into two equal parts.

Therefore, the total length of $BC$ can be calculated as:

$BC = BD + DC = 6 + 6 = 12$.

Thus, the length of side $BC$ is 12.

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

• Step 1: Identify the given information that $AD$ is the median making $BD = CD = 5$.
• Step 2: Use the triangle property that $BC = AC$ as per the problem’s isosceles condition.
• Step 3: Calculate $BC = BD + CD = 5 + 5 = 10$.
• Step 4: Since $BC = AC$, by isosceles property, conclude $AC = 10$.

Now, let's work through each step:
Step 1: With $AD$ as the median, it divides $BC$ into $BD = 5$ and $CD = 5$.
Step 2: The condition $BC = AC$ ensures the triangle is isosceles.
Step 3: Calculate $BC$ as $BD + CD = 5 + 5 = 10$.
Step 4: Since $BC = AC$, therefore, $AC = 10$.

Therefore, the length of $AC$ is $10$.

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, $58.5$.