Medians are drawn from point A and point B in this right triangle to divide segments BC and AC in half, respectively. The lengths of the medians are 6 and 2\sqrt{11} units, respectively. How many units are in the length of segment AB.
+26367 Medians are drawn from point A and point B in this right triangle to divide segments BC and AC in half, respectively. The lengths of the medians are 6 and \( 2\sqrt{11}\) units, respectively.
How many units are in the length of segment AB.
\(\text{Let $c = AB $} \\ \text{Let $a = BC $} \\ \text{Let $b = CA $} \)
\(\begin{array}{|lrcll|} \hline (1) & \left(\dfrac{1}{2}a \right)^2 + b^2 &=& 6^2 \\ (2) & a^2 + \left(\dfrac{1}{2}b \right)^2 &=& (2\sqrt{11})^2 \\ \hline (1) + (2): & \left(\dfrac{1}{2}a \right)^2 + b^2 + a^2 + \left(\dfrac{1}{2}b \right)^2 &=& 6^2 + (2\sqrt{11})^2 \\ & \dfrac{5}{4}a^2 + \dfrac{5}{4}b^2 &=& 36+44 \\ & \dfrac{5}{4}(a^2 + b^2) &=& 80 \quad & | \quad a^2+b^2=c^2 \\ & \dfrac{5}{4}c^2 &=& 80 \\ & c^2 &=& \dfrac{4\cdot 80}{5} \\ & c^2 &=& 4\cdot 16 \\ & c &=& 2\cdot 4 \\ & \mathbf{c} & \mathbf{=} & \mathbf{8} \\ \hline \end{array}\)
The length of segment AB = 8
