A rectangle $$ A B C D $$ is to be drawn on a centimetre grid such that $$ A $$ has coordinates $$ (-4,-2) $$ $$ B $$ has coordinates $$ (1,10) $$ $$ C $$ has coordinates $$ (19, a) $$ $$ D $$ has coordinates $$ (b, c) $$ Work out the value of $$ a $$, the value of $$ b $$ and the value of $$ c $$. (4 marks) $$ heta a= $$ $$ heta b= $$ $$ heta c= $$

Question

A rectangle $$ A B C D $$ is to be drawn on a centimetre grid such that $$ A $$ has coordinates $$ (-4,-2) $$ $$ B $$ has coordinates $$ (1,10) $$ $$ C $$ has coordinates $$ (19, a) $$ $$ D $$ has coordinates $$ (b, c) $$ Work out the value of $$ a $$, the value of $$ b $$ and the value of $$ c $$. (4 marks) $$ 	heta a= $$ $$ 	heta b= $$ $$ 	heta c= $$

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To determine the values of $$ a $$, $$ b $$, and $$ c $$ for rectangle $$ ABCD $$ with the given coordinates:

- $$ A = (-4, -2) $$
- $$ B = (1, 10) $$
- $$ C = (19, a) $$
- $$ D = (b, c) $$

**Step 1: Determine $$ b $$ and $$ c $$**

Since $$ ABCD $$ is a rectangle, the diagonals $$ AC $$ and $$ BD $$ must bisect each other. Therefore, the midpoints of $$ AC $$ and $$ BD $$ must be equal.

- **Midpoint of $$ AC $$:**
  $$
  \left( \frac{-4 + 19}{2}, \frac{-2 + a}{2} \right) = \left( \frac{15}{2}, \frac{a - 2}{2} \right)
  $$

- **Midpoint of $$ BD $$:**
  $$
  \left( \frac{1 + b}{2}, \frac{10 + c}{2} \right)
  $$

Setting the midpoints equal:
$$
\frac{1 + b}{2} = \frac{15}{2} \quad \Rightarrow \quad b = 14
$$
$$
\frac{10 + c}{2} = \frac{a - 2}{2} \quad \Rightarrow \quad c = a - 12
$$

**Step 2: Determine $$ a $$**

For $$ ABCD $$ to be a rectangle, adjacent sides must be perpendicular. Compute the vectors $$ \overrightarrow{AB} $$ and $$ \overrightarrow{AD} $$:

- $$ \overrightarrow{AB} = (1 - (-4), 10 - (-2)) = (5, 12) $$
- $$ \overrightarrow{AD} = (14 - (-4), c - (-2)) = (18, a - 10) $$

The dot product of these vectors must be zero:
$$
5 \cdot 18 + 12 \cdot (a - 10) = 0 \quad \Rightarrow \quad 90 + 12a - 120 = 0 \quad \Rightarrow \quad 12a = 30 \quad \Rightarrow \quad a = \frac{30}{12} = \frac{5}{2}
$$

**Step 3: Substitute $$ a $$ to find $$ c $$**
$$
c = a - 12 = \frac{5}{2} - 12 = \frac{5 - 24}{2} = \frac{-19}{2}
$$

**Final Values:**
$$
\theta a = \frac{5}{2}
$$
$$
\theta b = 14
$$
$$
\theta c = \frac{-19}{2}
$$

**Answer Summary:**
$$
\theta a = \frac{5}{2}, \quad \theta b = 14, \quad \theta c = \frac{-19}{2}
$$
$$ a = \frac{5}{2} $$, $$ b = 14 $$, $$ c = -\frac{19}{2} $$

A rectangle is to be drawn on a centimetre grid such that
has coordinates
has coordinates
has coordinates
has coordinates
Work out the value of , the value of and the value of .
(4 marks)

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