Question

14.
\frac{6\sqrt{5}}{5}
a
45°
b

Explanation:

Step1: Identify triangle type

This is a 45 - 45 - 90 right - triangle, so $a = b$.

Step2: Apply Pythagorean theorem

In a right - triangle, $a^{2}+b^{2}=c^{2}$. Since $a = b$, we have $2a^{2}=c^{2}$. Given $c=\frac{6\sqrt{5}}{5}$, then $2a^{2}=(\frac{6\sqrt{5}}{5})^{2}$.

Step3: Calculate $a^{2}$

$(\frac{6\sqrt{5}}{5})^{2}=\frac{36\times5}{25}=\frac{36}{5}$, so $2a^{2}=\frac{36}{5}$, and $a^{2}=\frac{18}{5}$.

Step4: Solve for $a$ (and $b$)

$a = b=\sqrt{\frac{18}{5}}=\frac{\sqrt{18}}{\sqrt{5}}=\frac{3\sqrt{2}}{\sqrt{5}}=\frac{3\sqrt{10}}{5}$.

Answer:

$a=\frac{3\sqrt{10}}{5}, b = \frac{3\sqrt{10}}{5}$