Question

what is the measure of angle b in the figure below?
a
3
b
3√2
c
30°
45°
60°
90°

Explanation:

Step1: Identify triangle type

The triangle is right - angled at \(A\), so it is a right - triangle. Let's denote the sides: \(AB = 3\), hypotenuse \(BC=3\sqrt{2}\), and let \(AC\) be the other leg.

Step2: Use trigonometric ratio or Pythagorean theorem

First, we can use the Pythagorean theorem to find \(AC\). In a right - triangle, \(AB^{2}+AC^{2}=BC^{2}\). Substituting the values: \(3^{2}+AC^{2}=(3\sqrt{2})^{2}\).
\(9 + AC^{2}=9\times2 = 18\)
\(AC^{2}=18 - 9=9\), so \(AC = 3\).

Since \(AB = AC = 3\) in a right - triangle, the triangle is an isosceles right - triangle. In an isosceles right - triangle, the non - right angles are equal and each measures \(45^{\circ}\). Angle \(B\) is one of the non - right angles, so \(\angle B = 45^{\circ}\).

We can also use the cosine ratio. \(\cos B=\frac{AB}{BC}\). Substituting \(AB = 3\) and \(BC = 3\sqrt{2}\), we get \(\cos B=\frac{3}{3\sqrt{2}}=\frac{1}{\sqrt{2}}\). We know that \(\cos45^{\circ}=\frac{1}{\sqrt{2}}\), so \(\angle B = 45^{\circ}\).

Answer:

\(45^{\circ}\) (corresponding to the option with \(45^{\circ}\))