Answer:
x = [tex]\sqrt{114}[/tex] , y = [tex]\sqrt{150}[/tex] , z = [tex]\sqrt{475}[/tex]
Step-by-step explanation:
In the larger right triangle the altitude is drawn from the right angle to the hypotenuse, dividing the hypotenuse into two segments.
To find the length of the altitude x , we can use the Geometric Mean Theorem , which states that the ratio of one segment to the altitude is equal to the ratio of the altitude to the other segment.
Here the segments are 6 and 25 - 6 = 19 , then
[tex]\frac{6}{x}[/tex] = [tex]\frac{x}{19}[/tex] ( cross multiply )
x² = 6 × 19 = 114 ( take square root of both sides )
[tex]\sqrt{x^2}[/tex] = [tex]\sqrt{114}[/tex]
x = [tex]\sqrt{114}[/tex]
To find y and z
Using Pythagoras' identity in the two smaller right triangles
• c² = a² + b² ( c is the hypotenuse and a, b the legs
let a = 6, b = x and c = y , then
y² = 6² + ( [tex]\sqrt{114}[/tex] )² = 36 + 114 = 150 ( take square root of both sides )
[tex]\sqrt{y^2}[/tex] = [tex]\sqrt{150}[/tex]
y = [tex]\sqrt{150}[/tex]
and
let a = 19 , b = x and c = z , then
z² = 19² + ( [tex]\sqrt{114}[/tex] )² = 361 + 114 = 475 ( take square root of both sides )
[tex]\sqrt{z^2}[/tex] = [tex]\sqrt{475}[/tex]
z = [tex]\sqrt{475}[/tex]