The point A is shown on the diagram - Junior Cycle Mathematics - Question Question 1 - 2012

The points are plotted as follows:

• B (2, 0)
• C (-4, -4)
• D (0, 4)
• E (-6, 0)
• F (4, -4)

To find the midpoint of segment DF, use the midpoint formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Here, D = (0, 4) and F = (4, -4).

Calculating: $M = \left( \frac{0 + 4}{2}, \frac{4 + (-4)}{2} \right) = \left( 2, 0 \right)$

The slope (m) is calculated using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ For points B (2, 0) and F (4, -4): $m = \frac{-4 - 0}{4 - 2} = \frac{-4}{2} = -2$

Using point B (2, 0) and the slope m = -2, we use the point-slope form: y - y_1 = m(x - x_1) Substituting: y - 0 = -2(x - 2) Expanding gives: y = -2x + 4.

For points C (-4, -4) and E (-6, 0): m = \frac{0 - (-4)}{-6 - (-4)} = \frac{4}{-2} = -2.

We can use the slope from the previous step. Using point C (-4, -4): y + 4 = -2(x + 4) Simplifying: y + 4 = -2x - 8, thus: 2x + y + 12 = 0.

The areas of triangles BCE and BCF can be calculated using the formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Given height is the difference in y-coordinates and base is the distance between points on the x-axis. After calculations:

• Area of ΔBCE = 8.
• Area of ΔBCF = 16. The ratio is therefore: $\text{Ratio} = \frac{1}{2}$ or 1:2.

Answer: yes. Reason: CFBE is a parallelogram and CB is a diagonal which divides the parallelogram into two congruent triangles. This follows by SSS or SAS argument.