The diagram below shows the circle k (not to scale) - Junior Cycle Mathematics - Question 12 - 2022

In triangle ABC, we have:

• AB = 10 cm
• AC = 8 cm
• BC = ?
  Using the cosine rule:
  $c^2 = a^2 + b^2 - 2ab imes ext{cos}(C)$
  We rewrite it as:
  To find angle C, we use AB and AC:
  Let |BC| = c:
  $c^2 = (10)^2 + (8)^2 - 2 imes 10 imes 8 imes ext{cos}(C)$

Assuming we calculate |BC|:
Using Pythagorean properties since [AB] is a diameter,
Angle C should ideally be a right angle. Thus:
$C = 90 ext{°}$

Next, we apply the sine rule to find angles A and B:
rac{AC}{ ext{sin}(A)} = rac{AB}{ ext{sin}(C)}
Solving gives:

• As angle C is 90°, angle A can further be derived or estimated using the right triangle characteristics:
  Based on the triangle angles sum rule, allowing us to approximate or determine angle B.

Finally, compared to angles A and B, after all calculations and derivations, estimate the smallest angle, leading to a minimum of around 36.87°, confirming through sine or cosine evaluations for narrowest angle extraction providing:
$ext{Smallest angle in } ABC ext{ is approximately } 36.87 ext{°}.$