Question

Any help would be appreciated. Thank you!

Answer 1

Answer:

Area $=62.5\sqrt{6}$ square units

$AB=5\sqrt{15}$ units

$BC=5\sqrt{10}$ units

Step-by-step explanation:

In a right triangle the altitude drawn to the hypotenuse is the geometric mean of the segments at which this altitude divides the hypotenuse.

So,

$BD^2=15\cdot 10\\ \\BD^2=150\\ \\BD=\sqrt{150}=5\sqrt{6}\ units$

a. The area of the triangle ABC is

$A_{ABC}=\dfrac{1}{2}\cdot BD\cdot AC=\dfrac{1}{2}\cdot 5\sqrt{6}\cdot (15+10)=\dfrac{125\sqrt{6}}{2}=62.5\sqrt{6}\ un^2.$

b. The legs of the right triangle are geometric means of the segment adjacent to this leg and the hypotenuse, so

$AB^2=AD\cdot AC=15\cdot 25\Rightarrow AB=5\sqrt{15}\ units\\ \\BC^2=CD\cdot AC=10\cdot 25\Rightarrow BC=5\sqrt{10}\ units$