a) We can actually use a linear equation for x, given by:
x*(3 + √5) = 10cm
With the solution x = 1.91cm
b) The longest side is:
√5*x = 4.27cm
You need to remember the Pythagorean's theorem, it says that the sum of the squares of the two shorter sides on a right triangle is equal to the square of the hypotenuse.
a) Here we know that the two shorter sides have a length:
x and 2x.
Then we have:
x^2 + (2x)^2 = H^2
x^2 + 4*x^2 = H^2
Where H is the hypotenuse, then we can rewrite:
5*x^2 = H^2
(√5)*x = H
Finally, we know that the perimeter of the triangle is 10 cm, then we have:
x + 2x + √5*x = 10cm
x*(1 + 2 + √5) = 10cm
x*(3 + √5) = 10cm
x = (10 cm)/(3 + √5) = 1.91cm
Notice that in this procedure we got a linear equation instead of the quadratic one we would got in point a (but the solution is the same, just a simpler approach.).
b) The length of the longest side is:
√5*x = √5*1.91cm = 4.27cm
If you want to learn more about right triangles, you can read:
https://brainly.com/question/2217700
