The shorter leg of a right triangle is 14 feet shorter than the longer leg. The hypotenuse is 26 feet. How long is each leg?
(X-14)^2+X^2 = 26^2
solve for x and x-14
a=shorter leg
b=longer leg
c=hypotenuse
a=b-14
c^2=a^2+b^2
26^2=(b-14)^2+b^2
676=b^2-2*b*14+14^2+b^2
676=2b^2-28b+196
676-196=2(b^2-14b)
480=2(b^2-14b) Divide with 2
240=b^2-14b
b^2-14b=240
b^2-14b+7^2=240+7^2
Becouse: (b-7)^2=b^2-14b+7^2
(b-7)^2=240+49
(b-7)^2=289
b-7=sqroot(289)
You have two solutions:
b-7=17 and
b-7=-17
Becouse sqroot(289)=+/- 17
First solution: b-7=17
b=17+7=24
Second solution:
b-7=-17
b=17+7=-10
leg of triangle can not be negative number so:
b=24
a=b-14=24-14=10
a=10 b=24
c=sqroot(a^2+b^2)
c=sqroot(10^2+24^2)
c=sqroot(100+576)
c=sqroot(676)
c=26
Ah, we have ourselves a classic right triangle conundrum! Let's do some mathematical clowning around with this one.
Let's call the longer leg "x" feet, just to give it a little personal touch. According to the problem, we know that the shorter leg is 14 feet shorter than the longer leg. So, the shorter leg measures (x - 14) feet.
Now, as any triangular circus clown will tell you, the Pythagorean theorem saves us from mathematical mayhem. According to the theorem, the sum of the squares of the two shorter sides equals the square of the hypotenuse.
So, we have (x - 14)^2 + x^2 = 26^2.
Squaring, summing, and juggling, we end up with x^2 - 28x + 196 + x^2 = 676.
Combining like terms here and there, we have 2x^2 - 28x + 196 - 676 = 0.
Simplifying clownishly, we have 2x^2 - 28x - 480 = 0.
Factoring this equation, we have (2x + 10)(x - 24) = 0.
Therefore, we have two potential solutions. Either 2x + 10 = 0, or x - 24 = 0.
But let's face it, negative numbers and clowns don't mix well, so we'll throw out the negative solution.
Thus, we have x - 24 = 0, which means x = 24.
Substituting back, the longer leg is 24 feet, and since the shorter leg is 14 feet shorter, it must be 24 - 14 = 10 feet.
So there you have it— the shorter leg is 10 feet and the longer leg is 24 feet, making for quite the dynamic clown duo!
Let's call the longer leg of the right triangle "x" feet.
According to the problem, the shorter leg is 14 feet shorter than the longer leg. So, the length of the shorter leg would be (x - 14) feet.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Using this theorem, we can write the equation:
(x - 14)^2 + x^2 = 26^2
Expanding the equation:
x^2 - 28x + 196 + x^2 = 676
Combining like terms:
2x^2 - 28x + 196 = 676
Subtracting 676 from both sides:
2x^2 - 28x - 480 = 0
Dividing both sides by 2 to simplify the equation:
x^2 - 14x - 240 = 0
Now we can factor the quadratic equation:
(x - 24)(x + 10) = 0
Setting each factor equal to zero and solving for x:
x - 24 = 0 or x + 10 = 0
If x - 24 = 0, then x = 24.
If x + 10 = 0, then x = -10.
Since the length of a side cannot be negative, we can disregard the solution x = -10.
Therefore, the longer leg of the right triangle is 24 feet.
Now, let's find the length of the shorter leg:
Shorter leg = x - 14 = 24 - 14 = 10 feet.
So, the longer leg is 24 feet, and the shorter leg is 10 feet.
To find the lengths of the legs of the right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's assume that the longer leg of the triangle is x feet. Since the shorter leg is described as being 14 feet shorter than the longer leg, we can represent it as (x - 14) feet.
Applying the Pythagorean theorem, we have the following equation:
(x - 14)^2 + x^2 = 26^2
Simplifying the equation, we get:
x^2 - 28x + 196 + x^2 = 676
Combine like terms:
2x^2 - 28x + 196 - 676 = 0
2x^2 - 28x - 480 = 0
Divide the equation by 2 to simplify it further:
x^2 - 14x - 240 = 0
Now, we can solve this quadratic equation to find the value of x. We can either factorize the quadratic equation or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -14, and c = -240. Substituting the values into the formula:
x = (-(-14) ± √((-14)^2 - 4(1)(-240))) / (2(1))
Simplifying further:
x = (14 ± √(196 + 960)) / 2
x = (14 ± √1156) / 2
x = (14 ± 34) / 2
Now we have two possible values for x:
x = (14 + 34) / 2 = 48 / 2 = 24
x = (14 - 34) / 2 = -20 / 2 = -10
Since lengths cannot be negative, we discard the negative value (-10). Thus, the longer leg of the right triangle is 24 feet.
To find the length of the shorter leg, which is 14 feet shorter than the longer leg, we subtract 14 from 24:
Shorter leg = 24 - 14 = 10 feet
Therefore, the longer leg of the right triangle is 24 feet, and the shorter leg is 10 feet.