Answer:
9.6 mi
Step-by-step explanation:
in right triangle ABC,
AC = 16 mi
BC = 24 mi
If AM is the median, then CM = MB = 12 mi
Consider triangles CAM and EBM. In these triangles,
- angles CMA and EMB are congruent as vertical angles;
- angles ACM and MEB are congruent as right angles.
So, triangles CAM and EBM are similar by AA postulate.
Similar triangles have proportional corresponding sides, so
[tex]\dfrac{CA}{EB}=\dfrac{CM}{EM}\\ \\\dfrac{16}{EB}=\dfrac{12}{EM}\\ \\EM=\dfrac{3}{4}EB[/tex]
By the Pythagorean theorem,
[tex]MB^2=EB^2+EM^2\\ \\12^2=EB^2+\left(\dfrac{3}{4}EB\right)^2\\ \\144=EB^2+\dfrac{9}{16}EB^2\\ \\EB^2\left(1+\dfrac{9}{16}\right)=144\\ \\EB^2\cdot \dfrac{25}{16}=144\\ \\EB^2=144\cdot \dfrac{16}{25}\\ \\EB=12\cdot \dfrac{4}{5}\\ \\EB=\dfrac{48}{5}\\ \\EB=9.6\ mi[/tex]
