Question

d) a 6.0 cm b (right angle) 5.0 cm c 2.0 cm d (right angle) i
statement reason

Explanation:

Step1: Identify Similar Triangles

Triangles \(ABC\) and \(DEC\) (or \(D C i\) - let's assume the small triangle is \( \triangle D C i\) with right angle at \(D\)) are similar by AA (Angle - Angle) similarity criterion. Both have a right angle (\(\angle B = \angle D = 90^\circ\)) and share \(\angle ACB=\angle D Ci\) (vertical angles). So, the ratios of corresponding sides are equal.
Let the height of the small triangle (length \(i\)) be \(h\). Then, \(\frac{AB}{h}=\frac{BC}{CD}\)

Step2: Substitute Values

We know \(AB = 6.0\space cm\), \(BC = 5.0\space cm\), \(CD = 2.0\space cm\). Substituting into the proportion:
\(\frac{6.0}{h}=\frac{5.0}{2.0}\)

Step3: Solve for \(h\)

Cross - multiply: \(5.0\times h=6.0\times2.0\)
\(5h = 12\)
\(h=\frac{12}{5}=2.4\space cm\)

Answer:

The length of \(i\) is \(2.4\space cm\)