💥 Q 76 Prep 3 Geometry Question Bank T1

In the opposite figure: The point C is the midpoint of \( \overrightarrow{AB} \) where \( C (4, 3) \).

1. Find the coordinates of each of: O, A and B

2. Find the length of each of: \( \overrightarrow{OA} \), \( \overrightarrow{OB} \), \( \overrightarrow{CA} \), \( \overrightarrow{CB} \) and \( \overrightarrow{CO} \)

3. Find the slope of each of: \( \overrightarrow{AB} \), \( \overrightarrow{OC} \), \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \)

4. Find the equation of each of: \( \overrightarrow{AB} \) and \( \overrightarrow{CO} \)

1. Coordinates:
\( O = (0, 0) \)
Let \( A = (x, 0) \), \( B = (0, y) \)
Since C is midpoint: \( C = \left(\frac{x+0}{2}, \frac{0+y}{2}\right) = (4, 3) \)
∴ \( \frac{x}{2} = 4 \) ⇒ \( x = 8 \)
\( \frac{y}{2} = 3 \) ⇒ \( y = 6 \)
∴ \( A (8, 0) \), \( B (0, 6) \)

2. Lengths:
\( \overrightarrow{OA} = 8 \) length units
\( \overrightarrow{OB} = 6 \) length units
\( \overrightarrow{CA} = \sqrt{(8-4)^2 + (0-3)^2} = 5 \) length units
\( \overrightarrow{CB} = \sqrt{(0-4)^2 + (6-3)^2} = 5 \) length units
\( \overrightarrow{CO} = \sqrt{(4-0)^2 + (3-0)^2} = 5 \) length units

3. Slopes:
Slope of \( \overrightarrow{AB} = \frac{6-0}{0-8} = -\frac{3}{4} \)
Slope of \( \overrightarrow{OC} = \frac{3-0}{4-0} = \frac{3}{4} \)
Slope of \( \overrightarrow{OA} = 0 \) (horizontal line)
Slope of \( \overrightarrow{OB} \) is undefined (vertical line)

4. Equations:
Equation of \( \overrightarrow{AB}: y = -\frac{3}{4}x + 6 \)
Equation of \( \overrightarrow{CO}: y = \frac{3}{4}x \)

MR/Nasef.N