Complete the following theorem: If a line divides two sides of a triangle in the same proportion, then the line is .. - NSC Technical Mathematics - Question 9 - 2019 - Paper 2

Since triangles ΔODF and ΔOBC are similar (by AA similarity), we can establish the ratios of their corresponding sides:

• Given OD : OB = 3 : 5, therefore, the ratio is:

$BC : DF = 10 : 6 = 5 : 3.$

Thus, the ratio of BC to DF is 5 : 3.

To find EG, we employ the proportional relationship established earlier:

• Given OD = 6 units and using the ratio, we find:

OE = \frac{6}{5} \cdot 3 = 3.6 \text{ units.}$$ Thus, we compute: $$EG = OE - OD = 6 - 3.6 = 2.4 \text{ units.}$$

To find the ratio of the areas, we can use the formula for the area of a triangle:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}.$

For triangle OBG:

• Base = OG, Height = OB, Thus:

$\text{Area} (OBG) = \frac{1}{2} (6)(3.6) = 10.8.$

For triangle ODE:

• Base = OE, Height = OD = 3, Thus:

$\text{Area} (ODE) = \frac{1}{2} (4)(3) = 6.$

Thus, the numerical value of the ratio is: $\text{Area} △ OBG / Area △ ODE = \frac{10.8}{6} = 1.8.$

To establish the similarity of triangles △ DOE and △ BOG:

1. Common Angle:

   • We note that (\angle DOE = \angle BOG).

2. Right Angles:

   • Both (\angle OED = 90^\circ) and (\angle OBG = 90^\circ).

Thus, by AA criterion, triangles are similar:

• Therefore, △ DOE || △ BOG, as stated.