Question

Solve for x and y

Both questions please ❤️​

Answer 1

1.
`x=18, y=6√(10)`

2. x=108, y=180

Explanation:

The height of the right triangle drawn to the hypotenuse is the geometric mean of two segments of the hypotenuse (two legs' projections)

1. Use this property, so

`CD^2=AD\cdot BD\\ \\6^2=2\cdot x\\ \\36=2x\\ \\x=18`

Consider right triangle BCD. By the Pythagorean theorem,

`BC^2=BD^2+CD^2\\ \\y^2=18^2+6^2\\ \\y^2=324+36\\ \\y^2=360\\ \\y=√(360)=6√(10)`

2. Consider right triangleGM*. By the Pythagorean theorem,

`G*^2=GM^2+M*^2\\ \\240^2=GM^2+192^2\\ \\GM^2=240^2-192^2=(240-192)(240+192)=48\cdot 432=144^2\\ \\GM=144`

Use this property to find x:

`GM^2=192\cdot x\\ \\144^2=192\cdot x\\ \\x=(144^2)/(192)=108`

Consider right triangle GMCup. By the Pythagorean theorem,

`y^2=x^2+GM^2\\ \\y^2=108^2+144^2\\ \\y^2=32,400\\ \\y=180`