Respuesta :
Answer: hypotenuse = [tex]c^{2} + b^{2}[/tex]
Step-by-step explanation: Pythagorean theorem states that square of hypotenuse (h) equals the sum of squares of each side ([tex]s_{1},s_{2}[/tex]) of the right triangle, .i.e.:
[tex]h^{2} = s_{1}^{2} + s_{2}^{2}[/tex]
In this question:
[tex]s_{1}[/tex] = [tex]c^{2}-b^{2}[/tex]
[tex]s_{2} =[/tex] 2bc
Substituing and taking square root to find hypotenuse:
[tex]h=\sqrt{(c^{2}-b^{2})^{2}+(2bc)^{2}}[/tex]
Calculating:
[tex]h=\sqrt{c^{4}+b^{4}-2b^{2}c^{2}+(4b^{2}c^{2})}[/tex]
[tex]h=\sqrt{c^{4}+b^{4}+2b^{2}c^{2}}[/tex]
[tex]c^{4}+b^{4}+2b^{2}c^{2}[/tex] = [tex](c^{2}+b^{2})^{2}[/tex], then:
[tex]h=\sqrt{(c^{2}+b^{2})^{2}}[/tex]
[tex]h=(c^{2}+b^{2})[/tex]
Hypotenuse for the right-angled triangle is [tex]h=(c^{2}+b^{2})[/tex] units
The expression for the length of the hypotenuse is [tex]c^2+ b^2 \ units[/tex].
Given that,
A right-angled triangle has shorter side lengths exactly [tex]c^{2} - b^{2}[/tex] and 2bc units respectively,
Where b and c are positive real numbers such that c is greater than b.
We have to determine,
An exact expression for the length of the hypotenuse?
According to the question,
The Pythagoras theorem states that the sum of the hypotenuse in the right-angled triangle is equal to the sum of the square of the other two sides.
A right-angled triangle has shorter side lengths exactly [tex]c^{2} - b^{2}[/tex] and 2bc units respectively,
Where b and c are positive real numbers such that c is greater than b.
Therefore,
The expression for the length of the hypotenuse is,
[tex]\rm (Hypotenuse)^2 = (c^2-b^2)^2+ (2bc)^2\\\\(Hypotenuse)^2 = c^4+ b^4 - 2c^2b^2 + 4c^2b^2\\\\(Hypotenuse)^2 = c^4+ b^4 +2c^2b^2 \\\\Hypotenuse = \sqrt{c^4+ b^4 +2c^2b^2}\\\\\ Hypotenuse = \sqrt{(c^2+ b^2)^2}\\\\ Hypotenuse = c^2+b^2 \ units[/tex]
Hence, The required expression for the length of the hypotenuse is [tex]c^2+ b^2 \ units[/tex].
For more details refer to the link.
https://brainly.com/question/9214495
