5 (a) In the following figure, triangles ABE and ACD are similar. It is given that AB = 6cm, BE = 8cm, CD = 12cm and BE is parallel to CD. Calculate the length of BC. [3] (b) Given that ( -3 5 ) ( 2 3 ) = ( 9 38 ), find the value of x. [3] (c) The diagram shows triangle ABC drawn on the XOY-plane. (i) Write the coordinates of A, B and C. [3]

Solve and explain

Question

5 (a) In the following figure, triangles ABE and ACD are similar. It is given that AB = 6cm, BE = 8cm, CD = 12cm and BE is parallel to CD. Calculate the length of BC. [3] (b) Given that ( -3 5 ) ( 2 3 ) = ( 9 38 ), find the value of x. [3] (c) The diagram shows triangle ABC drawn on the XOY-plane. (i) Write the coordinates of A, B and C. [3]

Accepted Answer

Let's solve each part step by step.

### Part (a)

**Problem:**
Triangles ABE and ACD are similar. Given:
- $ AB = 6 \, \text{cm} $
- $ BE = 8 \, \text{cm} $
- $ CD = 12 \, \text{cm} $

We need to find the length of $ BC $.

**Solution:**
Since triangles ABE and ACD are similar and $ BE \parallel CD $, the sides are proportional:

$$
\frac{AB}{AC} = \frac{BE}{CD}
$$

Substitute the known values:

$$
\frac{6}{AC} = \frac{8}{12}
$$

Simplify the right side:

$$
\frac{6}{AC} = \frac{2}{3}
$$

Cross-multiply to solve for $ AC $:

$$
2 \times AC = 6 \times 3
$$
$$
2 \times AC = 18
$$
$$
AC = 9
$$

Now, $ AC = AB + BC $, so:

$$
9 = 6 + BC
$$
$$
BC = 3 \, \text{cm}
$$

### Part (b)

**Problem:**
Given the matrix equation:

$$
\begin{pmatrix} -3 & 5 \ 2x & 5x \end{pmatrix} \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 9 \ 38 \end{pmatrix}
$$

Find the value of $ x $.

**Solution:**
Multiply the matrices:

$$
\begin{pmatrix} -3 \times 2 + 5 \times 3 \ 2x \times 2 + 5x \times 3 \end{pmatrix} = \begin{pmatrix} 9 \ 38 \end{pmatrix}
$$

Calculate the first row:

$$
-6 + 15 = 9
$$

This is correct. Now calculate the second row:

$$
4x + 15x = 38
$$
$$
19x = 38
$$
$$
x = 2
$$

### Part (c)

**Problem:**
Write the coordinates of points A, B, and C from the graph.

**Solution:**

From the graph, the coordinates are:
- $ A = (3, 6) $
- $ B = (2, 2) $
- $ C = (6, 0) $

These solutions address all parts of the problem.