Question

11 find the length of $overline{xz}$.

Explanation:

Step1: Identify triangle type & sides

We have triangle $XYZ$ (central triangle) with $XY = ZY = 6$, $\angle XYZ = 109^\circ$. Use the Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$

Step2: Substitute values into formula

$$ XZ^2 = 6^2 + 6^2 - 2(6)(6)\cos(109^\circ) $$

Calculate each term: $6^2=36$, $2(6)(6)=72$, $\cos(109^\circ)\approx-0.3256$

$$ XZ^2 = 36 + 36 - 72(-0.3256) $$

Step3: Compute intermediate results

$$ XZ^2 = 72 + 23.4432 = 95.4432 $$

Step4: Solve for $XZ$

Take square root of both sides:

$$ XZ = \sqrt{95.4432} $$

Answer:

$\approx 9.77$