Question 1

We will use the Basic Proportionality Theorem (BPT) because DE is parallel to BC in both cases.

Step 1 — Find EC in (i)

Let's look at the first triangle. We are given $AD = \mathbf{1.5 \text{ cm}}$. We are given $DB = \mathbf{3 \text{ cm}}$. We are given $AE = \mathbf{1 \text{ cm}}$. We need to find the length of EC. Let's call EC as $x$. Since DE is parallel to BC, we can apply the BPT. The theorem states that the ratio $AD/DB$ equals $AE/EC$.

$\frac{AD}{DB} = \frac{AE}{EC}$

Let's substitute the given values into the equation.

$\frac{1.5}{3} = \frac{1}{x}$

Now, let's cross-multiply to solve for $x$.

$1.5 \times x = 3 \times 1$

$1.5x = 3$

Let's divide both sides by 1.5.

$x = \frac{3}{1.5}$

$x = 2$

$\boxed{EC = 2 \text{ cm}}$

Step 2 — Find AD in (ii)

Let's look at the second triangle. We are given $DB = \mathbf{7.2 \text{ cm}}$. We are given $AE = \mathbf{1.8 \text{ cm}}$. We are given $EC = \mathbf{5.4 \text{ cm}}$. We need to find the length of AD. Let's call AD as $x$. Again, DE is parallel to BC, so we use the BPT. The theorem states that the ratio $AD/DB$ equals $AE/EC$.

$\frac{AD}{DB} = \frac{AE}{EC}$

Let's substitute the given values into the equation.

$\frac{x}{7.2} = \frac{1.8}{5.4}$

Now, let's solve for $x$.

$x = \frac{1.8 \times 7.2}{5.4}$

We can simplify the fraction $1.8/5.4$ first.

$\frac{1.8}{5.4} = \frac{18}{54} = \frac{1}{3}$

Let's substitute this simplified fraction back into the equation for $x$.

$x = \frac{1}{3} \times 7.2$

$x = \frac{7.2}{3}$

$x = 2.4$

$\boxed{AD = 2.4 \text{ cm}}$

Answer

(i) $EC = 2 \text{ cm}$ (ii) $AD = 2.4 \text{ cm}$