A piece of wire 40cm is bent to form aright-angle triangle whose hypotenuse is 17cm long.Find the lengths of two sides of the triangle
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To find the lengths of the two sides of the right-angle triangle, we can use the Pythagorean theorem, which states that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's assume that the two sides of the triangle are x cm and y cm, where x and y are the lengths of the two sides. So, we have:
x + y + 17 = 40 (since the wire is 40 cm long)
x^2 + y^2 = 17^2 (using the Pythagorean theorem)
Now, we can solve these two equations to find the values of x and y.
From the first equation, we can rewrite it as:
x + y = 40 - 17
x + y = 23
Now, we can rearrange the second equation:
x^2 + y^2 = 289
From the first equation, we have x = 23 - y. Substituting this into the second equation, we get:
(23 - y)^2 + y^2 = 289
529 - 46y + y^2 + y^2 = 289
2y^2 - 46y + 529 - 289 = 0
2y^2 - 46y + 240 = 0
y^2 - 23y + 120 = 0
(y - 8)(y - 15) = 0
From this quadratic equation, we have two possibilities:
y - 8 = 0
y = 8
y - 15 = 0
y = 15
So, the lengths of the two sides of the triangle are 8 cm and 15 cm.
The other two sides have a total length of 40-17 = 23, so if one side is x cm, then
x^2 + (23-x)^2 = 17^2
now finish it off.
By the way, you could save some calculation if you learned a few of the basic Pythagorean triples.