2. $$ A B C D $$ is a parallelogram such that $$ A(5,2) $$, $$ B(-2,-3) $$ and $$ \overrightarrow{A C}=\binom{1}{-3} $$ Find (i) the coordinates of $$ C $$ and $$ D $$; (ii) $$ \overrightarrow{B C} $$.

Question

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

## Given:
- **Point A:** $$ A(5, 2) $$
- **Point B:** $$ B(-2, -3) $$
- **Vector $$ \overrightarrow{AC} $$:** $$ \binom{1}{-3} $$

## Part (i): Find the Coordinates of $$ C $$ and $$ D $$

### Finding Point $$ C $$:
The vector $$ \overrightarrow{AC} $$ represents the displacement from point $$ A $$ to point $$ C $$. Mathematically, it's defined as:
$$
\overrightarrow{AC} = C - A
$$
Given $$ \overrightarrow{AC} = \binom{1}{-3} $$ and $$ A(5, 2) $$, we can find $$ C $$ as follows:
$$
C = A + \overrightarrow{AC} = (5, 2) + \left(1, -3\right) = (5 + 1, 2 - 3) = (6, -1)
$$

### Finding Point $$ D $$:
In a parallelogram $$ ABCD $$, the diagonals bisect each other. Therefore, the midpoint of diagonal $$ AC $$ is the same as the midpoint of diagonal $$ BD $$.

1. **Find the Midpoint of $$ AC $$:**
   $$
   \text{Midpoint}_{AC} = \left( \frac{5 + 6}{2}, \frac{2 + (-1)}{2} \right) = \left( \frac{11}{2}, \frac{1}{2} \right)
   $$

2. **Let $$ D = (x, y) $$. Find $$ D $$ using the midpoint formula for $$ BD $$:**
   $$
   \text{Midpoint}_{BD} = \left( \frac{-2 + x}{2}, \frac{-3 + y}{2} \right) = \left( \frac{11}{2}, \frac{1}{2} \right)
   $$
   
   Equate the corresponding components:
   $$
   \frac{-2 + x}{2} = \frac{11}{2} \quad \Rightarrow \quad -2 + x = 11 \quad \Rightarrow \quad x = 13
   $$
   $$
   \frac{-3 + y}{2} = \frac{1}{2} \quad \Rightarrow \quad -3 + y = 1 \quad \Rightarrow \quad y = 4
   $$
   
   So, $$ D = (13, 4) $$.

### Summary for Part (i):
- **Point $$ C $$:** $$ (6, -1) $$
- **Point $$ D $$:** $$ (13, 4) $$

## Part (ii): Find Vector $$ \overrightarrow{BC} $$

Vector $$ \overrightarrow{BC} $$ represents the displacement from point $$ B $$ to point $$ C $$:
$$
\overrightarrow{BC} = C - B
$$
Given:
- $$ B(-2, -3) $$
- $$ C(6, -1) $$

Calculate $$ \overrightarrow{BC} $$:
$$
\overrightarrow{BC} = (6 - (-2), -1 - (-3)) = (8, 2)
$$

### Summary for Part (ii):
- **Vector $$ \overrightarrow{BC} $$:** $$ \binom{8}{2} $$

---

**Final Answer:**

**(i)** The coordinates are $$ C(6,\ -1) $$ and $$ D(13,\ 4) $$;

**(ii)** The vector $$ \overrightarrow{BC} $$ is $$ \binom{8}{2} $$.
**(i)** Point $$ C $$ is at $$ (6, -1) $$ and point $$ D $$ is at $$ (13, 4) $$. **(ii)** The vector $$ \overrightarrow{BC} $$ is $$ \binom{8}{2} $$.

2. \( A B C D \) is a parallelogram such that \( A(5,2) \), \( B(-2,-3) \) and \( \overrightarrow{A C}=\binom{1}{-3} \) Find (i) the coordinates of \( C \) and \( D \); (ii) \( \overrightarrow{B C} \).

Real Tutor Solution

Answer

Solution

Mind Expander

Related Questions

Latest Geometry Questions