Given:
The figure of a triangle.
To find:
The value of x.
Solution:
According to the triangle proportionality theorem, if a line segment divides the two sides of a triangle and parallel to the third side of the triangle, the it divides the two inclined sides proportionally.
Using triangle proportionality theorem, we get
![\dfrac{2x}{80}=\dfrac{27}{60}](https://tex.z-dn.net/?f=%5Cdfrac%7B2x%7D%7B80%7D%3D%5Cdfrac%7B27%7D%7B60%7D)
![\dfrac{x}{40}=\dfrac{9}{20}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%7D%7B40%7D%3D%5Cdfrac%7B9%7D%7B20%7D)
Multiply both sides by 40.
![x=\dfrac{9}{20}\times 40](https://tex.z-dn.net/?f=x%3D%5Cdfrac%7B9%7D%7B20%7D%5Ctimes%2040)
![x=9\times 2](https://tex.z-dn.net/?f=x%3D9%5Ctimes%202)
![x=18](https://tex.z-dn.net/?f=x%3D18)
Therefore, the value of x is 18.
In interval notation, it would be (-inf., 2)
Longer leg = x + 4
Shorter leg = x
Hypotenuse = x + 8
Using Pythagorean Theorem:
a^2 + b^2 = c^2
(x + 4)^2 + x^2 = (x + 8)^2
x^2 + 8x + 16 + x^2 = x^2 + 16x + 64
2x^2 + 8x + 16 = x^2 + 16x + 64
2x^2 +8x = x^2 + 16x + 48
2x^2 - 8x = x^2 + 48
x^2 - 8x = 48
x^2 - 8x - 48 = 0
You can complete the square from here or use the quadratic formula.
Completing the square:
x^2 - 8x = 48
x^2 - 8x + (-8/2)^2 = 48 + (-8/2)^2
x^2 - 8x + 16 = 48 + 16
(x - 4)(x - 4) = 64 or (x - 4)^2 = 64
x - 4 = +√64 OR x - 4 = -√64
x - 4 = +8 OR x - 4 = -8
x = 12 OR x = -4
However, you can't use negative 4 as a length because your length needs to be a positive.
So x will be 12.
Shorter leg: 12
Longer leg: 12 + 4
Hypotenuse: 12 + 8
The midpoint is the average of the endpoint coords.
If (x,y) is F, then -6=(-14+x)/2 and 7=(13+y)/2.
-14+x=-12 so x=2, and 14=13+y, so y=1 and F is (2,1).
so the answer is (2,1)