Answer:
The formula for the area of the triangle is:
![\displaystyle A=24x^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%3D24x%5E2)
When <em>x</em> = 3, the area is 216 square centimeters.
When <em>x </em>= 2.4, the area is 138.24 square centimeters.
And when <em>x</em> = 5, the hypotenuse of the triangle is 50 centimeters.
Step-by-step explanation:
We are given a right triangle with a height of 8<em>x</em> and a base of 6<em>x</em>.
The area of a triangle is given by the formula:
![\displaystyle A=\frac{1}{2}bh](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%3D%5Cfrac%7B1%7D%7B2%7Dbh)
So, by substitution:
![\displaystyle A=\frac{1}{2}(8x)(6x)=24x^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%3D%5Cfrac%7B1%7D%7B2%7D%288x%29%286x%29%3D24x%5E2)
When <em>x </em>= 3, the area will be:
![\displaystyle A(3)=24(3)^2=216\text{ cm}^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%283%29%3D24%283%29%5E2%3D216%5Ctext%7B%20cm%7D%5E2)
And when <em>x</em> = 2.4, the area will be:
![A(2.4)=24(2.4)^2=138.24\text{ cm}^2](https://tex.z-dn.net/?f=A%282.4%29%3D24%282.4%29%5E2%3D138.24%5Ctext%7B%20cm%7D%5E2)
When <em>x </em>= 5, the height will be 8(5) = 40 cm and the base will be 6(5) = 30 cm.
By the Pythagorean Theorem:
![a^2+b^2=c^2](https://tex.z-dn.net/?f=a%5E2%2Bb%5E2%3Dc%5E2)
So:
![(40)^2+(30)^2=c^2](https://tex.z-dn.net/?f=%2840%29%5E2%2B%2830%29%5E2%3Dc%5E2)
The hypotenuse will be:
![c=\sqrt{1600+900}=\sqrt{2500}=50\text{ cm}](https://tex.z-dn.net/?f=c%3D%5Csqrt%7B1600%2B900%7D%3D%5Csqrt%7B2500%7D%3D50%5Ctext%7B%20cm%7D)