Answer:
y = 32.4
Step-by-step explanation:
To find the value of y, we can use the Geometric Mean Theorem (Leg Rule).
The Geometric Mean Theorem (Leg Rule) states that if an altitude is drawn from the right angle of a right triangle to its hypotenuse, dividing it into two segments, the ratio of the length of the hypotenuse to one of the legs equals the ratio of that leg to the segment of the hypotenuse adjacent to it:
[tex]\boxed{\sf \dfrac{Hypotenuse}{Leg\:1}=\dfrac{Leg\:1}{Segment\;1}}\quad \sf and \quad \boxed{\sf \dfrac{Hypotenuse}{Leg\:2}=\dfrac{Leg\:2}{Segment\;2}}[/tex]
From inspection of the given right triangle:
- Hypotenuse = 12 + 27 = 39
- Leg 1 = x
- Segment 1 = 12
- Leg 2 = y
- Segment 2 = 27
Substitute the values of the hypotenuse, leg 2 and segment 2 into the second formula and solve for y:
[tex]\dfrac{39}{y}=\dfrac{y}{27}\\\\\\27 \cdot 39=y\cdot y\\\\\\y^2=1053\\\\\\y=\sqrt{1053}\\\\\\y=32.4499614791...\\\\\\y=32.4\; \sf (nearest\;tenth)[/tex]
Therefore, the value of y rounded to the nearest tenth is:
[tex]\LARGE\boxed{\boxed{y = 32.4}}[/tex]
