the longer legs of a right triangle is 20 cm more than twice the length of the shorter leg. The length of the hypotenuse is 22 cm more than twice the length of the shorter leg. Find the side lengths of the triangle
x^2 + (2x+20)^2 = (2x+22)^2
x^2+ 4 x^2 + 80 x + 400 = 4 x^2+88x + 484
x^2 - 8 x - 84 = 0
(x-14)(x+6) = 0
x = 14
2x+20 = 48
2x+22 = 50
Let's assume the shorter leg of the right triangle is "x" cm.
According to the given information, the longer leg is 20 cm more than twice the length of the shorter leg. So, the length of the longer leg would be 2x + 20 cm.
The hypotenuse is 22 cm more than twice the length of the shorter leg. Therefore, the length of the hypotenuse is 2x + 22 cm.
Using the Pythagorean theorem for a right triangle (a^2 + b^2 = c^2), we can set up the equation:
x^2 + (2x + 20)^2 = (2x + 22)^2
Now, let's solve this equation step-by-step:
Expand the squared terms:
x^2 + 4x^2 + 80x + 400 = 4x^2 + 88x + 484
Combine like terms:
x^2 + 80x + 400 = 88x + 484
Subtract 88x from both sides:
x^2 + 80x - 88x + 400 = 484
Combine like terms again:
x^2 - 8x + 400 = 484
Subtract 484 from both sides:
x^2 - 8x + 400 - 484 = 0
Simplify:
x^2 - 8x - 84 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation: a = 1, b = -8, c = -84
x = (-(-8) ± √((-8)^2 - 4(1)(-84))) / (2(1))
x = (8 ± √(64 + 336)) / 2
x = (8 ± √400) / 2
x = (8 ± 20) / 2
Now, we have two possible values for x:
1. When x = (8 + 20) / 2 = 28 / 2 = 14
2. When x = (8 - 20) / 2 = -12 / 2 = -6
Since we're talking about the length of a side, it cannot be negative. Therefore, the shorter leg of the right triangle is 14 cm.
Using this value, we can find the other lengths:
The longer leg = 2x + 20 = 2(14) + 20 = 28 + 20 = 48 cm
The hypotenuse = 2x + 22 = 2(14) + 22 = 28 + 22 = 50 cm
So, the side lengths of the triangle are:
Shorter leg = 14 cm
Longer leg = 48 cm
Hypotenuse = 50 cm
To solve this problem, let's assign variables to the shorter leg, longer leg, and hypotenuse.
Let x be the length of the shorter leg.
Then, the longer leg is 20 cm more than twice the length of the shorter leg, so it can be represented as 2x + 20.
Similarly, the hypotenuse is 22 cm more than twice the length of the shorter leg, so it can be represented as 2x + 22.
Now, we can use the Pythagorean theorem to set up an equation and solve for x. The Pythagorean theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
Using the given information, we have the equation:
x^2 + (2x + 20)^2 = (2x + 22)^2
Now, let's solve for x step by step.
Expanding the equation:
x^2 + (4x^2 + 80x + 400) = (4x^2 + 88x + 484)
Combining like terms:
5x^2 + 80x + 400 = 4x^2 + 88x + 484
Subtracting (4x^2 + 88x + 484) from both sides of the equation:
5x^2 + 80x + 400 - 4x^2 - 88x - 484 = 0
Simplifying:
x^2 - 8x - 84 = 0
Now, we can factor the quadratic equation:
(x - 12)(x + 7) = 0
Setting each factor to zero and solving for x:
x - 12 = 0 --> x = 12
x + 7 = 0 --> x = -7
We discard the negative value since we are dealing with side lengths, which cannot be negative.
Therefore, the shorter leg has a length of 12 cm.
Now, we can substitute this value of x back into our expressions for the longer leg and hypotenuse to find their lengths.
The longer leg = 2x + 20 = 2(12) + 20 = 44 cm
The hypotenuse = 2x + 22 = 2(12) + 22 = 46 cm
Hence, the side lengths of the right triangle are:
Shorter leg: 12 cm
Longer leg: 44 cm
Hypotenuse: 46 cm