Question

b) a 13 mm b b 5 mm c (with right angle at c)

Explanation:

Step1: Identify the triangle type

This is a right - triangle with hypotenuse \( AB = 13\space mm \) and one leg \( BC=5\space mm \), and we need to find the other leg \( AC = b \). We use the Pythagorean theorem, which states that for a right - triangle with legs \( a\) and \( b \) and hypotenuse \( c \), \( a^{2}+b^{2}=c^{2} \). In this triangle, let \( BC = a = 5\space mm \), \( AC=b \), and \( AB = c = 13\space mm \).

Step2: Apply the Pythagorean theorem

From \( a^{2}+b^{2}=c^{2} \), we can solve for \( b \):
\( b^{2}=c^{2}-a^{2} \)
Substitute \( a = 5\) and \( c = 13 \) into the formula:
\( b^{2}=13^{2}-5^{2} \)
\( b^{2}=169 - 25 \)
\( b^{2}=144 \)

Step3: Solve for \( b \)

Take the square root of both sides:
\( b=\sqrt{144}=12 \)

Answer:

\( b = 12\space mm \)