Answer:
[tex]AC= h = 3.2\ m[/tex]
[tex]AB= b = 1.1\ m[/tex]
[tex]BC= p = 3\ m[/tex]
Step-by-step explanation:
Let x be the length of the one side of a right angle triangle.
Given:
The hypotenuse of a right triangle is 1 meter longer than twice the length of one leg, and the length of the other leg measure 3 meters.
Length of one leg b = x
So length of the hypotenuse h = 2x + 1
Length of other leg p = 3 m
We need to find the length of each unknown side.
Solution.
From the figure h is the length of the hypotenuse AC and b, p is length of the legs AB and BC
Using Pythagoras theorem.
[tex](AC)^{2} = (AB)^{2}+(BC)^{2}[/tex]
[tex]h^{2} = (b)^{2}+(p)^{2}[/tex]
Substitute all given value in above equation and then simplify.
[tex](2x+1)^{2} = x^{2}+(3)^{2}[/tex]
[tex]4x^{2}+1+4x= x^{2}+9[/tex]
[tex]4x^{2}-x^{2}-9+1+4x= 0[/tex]
[tex]3x^{2}+4x-8 = 0[/tex]
Now, we first find the root of the above equation.
Use quadratic formula with [tex]a=3, b=4, c=-8[/tex].
[tex]x=\frac{-b\pm \sqrt{(b)^{2}-4ac}}{2a}[/tex]
Put a, b and c value in above equation.
[tex]x=\frac{-4\pm \sqrt{(4)^{2}-4(3)(-8)}}{2(3)}[/tex]
[tex]x=\frac{-4\pm \sqrt{16+96}}{6}[/tex]
[tex]x=\frac{-4\pm \sqrt{112}}{6}[/tex]
[tex]x=\frac{-4\pm 4\sqrt{7}}{6}[/tex]
For positive sign
x = 1.1 m
So the length of the hypotenuse [tex]h = 2x+1[/tex]
[tex]h = 2\times 1.1+1[/tex]
[tex]h = 2.2+1[/tex]
[tex]h = 3.2\ m[/tex]
Therefore, the length of the each side of the right angle triangle is given below.
[tex]AC= h = 3.2\ m[/tex]
[tex]AB= b = 1.1\ m[/tex]
[tex]BC= p = 3\ m[/tex]
