Question

Any help would be appreciated. Thank you!

Answer 1

Area
`=62.5√(6)` square units

`AB=5√(15)` units

`BC=5√(10)` units

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=√(150)=5√(6)\ units`

a. The area of the triangle ABC is

`A_(ABC)=(1)/(2)\cdot BD\cdot AC=(1)/(2)\cdot 5√(6)\cdot (15+10)=(125√(6))/(2)=62.5√(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√(15)\ units\\ \\BC^2=CD\cdot AC=10\cdot 25\Rightarrow BC=5√(10)\ units`