Question

The vertices of triangle ABC are (7, 5), (5, 7) and (-3, 3) respectively. If the mid-point of BC is D, then the measure of AD will be

• A. 4 units
• B. 5 units
• C. 6 units
• D. 7 units

Hint:

When using the distance formula, if you notice that either the x-coordinates or the y-coordinates are the same, the distance is simply the absolute difference of the other coordinates. Here, for A(7, 5) and D(1, 5), the y-coordinates are the same. The distance is just \( |7 - 1| = 6 \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves two main concepts from coordinate geometry: finding the midpoint of a line segment and finding the distance between two points. We first need to find the coordinates of point D, and then calculate the length of the line segment AD.
Step 2: Key Formula or Approach:
1. Midpoint Formula: The coordinates of the midpoint of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) are \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
2. Distance Formula: The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Step 3: Detailed Explanation:
The vertices are A(7, 5), B(5, 7), and C(-3, 3).
First, find the coordinates of D, the midpoint of BC.
Using the midpoint formula with B(5, 7) and C(-3, 3):
\[ D = \left( \frac{5 + (-3)}{2}, \frac{7 + 3}{2} \right) \] \[ D = \left( \frac{2}{2}, \frac{10}{2} \right) \] \[ D = (1, 5) \] Next, find the length of the median AD using the distance formula with A(7, 5) and D(1, 5).
\[ AD = \sqrt{(1 - 7)^2 + (5 - 5)^2} \] \[ AD = \sqrt{(-6)^2 + (0)^2} \] \[ AD = \sqrt{36 + 0} \] \[ AD = \sqrt{36} \] \[ AD = 6 \] Step 4: Final Answer:
The measure of AD is 6 units.