If the coordinates of midpoints of ` AB and AC of Δ ` A B C` are (3,5) and (-3,-3) respectively, then write the length of side BC.

Options

A. 10
B. 15
C. 20
D. 30

Explanation

To find the length of side BC of triangle ABC given the coordinates of the midpoints of sides AB and AC, we can follow these steps:

### Step 1: Identify the coordinates of the midpoints
Let the midpoint D of side AB have coordinates (3, 5) and the midpoint E of side AC have coordinates (-3, -3).

### Step 2: Use the distance formula to find the length of DE
The distance formula between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, we will apply this formula to find the distance DE.

### Step 3: Substitute the coordinates into the distance formula
Substituting the coordinates of D (3, 5) and E (-3, -3) into the distance formula:
\[ DE = \sqrt{((-3) - 3)^2 + ((-3) - 5)^2} \]

### Step 4: Calculate the differences
Calculating the differences:
- For x-coordinates:
\[ -3 - 3 = -6 \]
- For y-coordinates:
\[ -3 - 5 = -8 \]

### Step 5: Square the differences
Now, we square the differences:
\[ DE = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} \]

### Step 6: Simplify the expression
Simplifying the expression:
\[ DE = \sqrt{100} = 10 \]

### Step 7: Use the midpoint theorem to find BC
According to the midpoint theorem, the length of side BC is twice the length of DE:
\[ BC = 2 \times DE = 2 \times 10 = 20 \]

### Final Answer
Thus, the length of side BC is 20 units.

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