Question

find ( mangle k ).
( mangle k=square^{circ} )

Explanation:

Step1: Identify the triangle type

This is a right - triangle with right angle at \(I\), hypotenuse \(KJ = 2\sqrt{33}\) and one of the legs \(KI=3\sqrt{11}\). We can use trigonometric ratios to find the angle \(K\). Let's recall the cosine ratio: \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)

Step2: Apply the cosine formula

For angle \(K\), the adjacent side to angle \(K\) is \(KI = 3\sqrt{11}\) and the hypotenuse is \(KJ=2\sqrt{33}\)

So, \(\cos(\angle K)=\frac{KI}{KJ}=\frac{3\sqrt{11}}{2\sqrt{33}}\)

Simplify the fraction: \(\frac{3\sqrt{11}}{2\sqrt{3\times11}}=\frac{3}{2\sqrt{3}}=\frac{\sqrt{3}}{2}\) (rationalizing the denominator: \(\frac{3}{2\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{3\sqrt{3}}{2\times3}=\frac{\sqrt{3}}{2}\))

Step3: Find the angle

We know that \(\cos\theta=\frac{\sqrt{3}}{2}\) and we are looking for an acute angle (since it's a right - triangle and angle \(K\) is acute). We know that \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\), so \(m\angle K = 30^{\circ}\)

Answer:

\(30\)