The hypotenuse of a right triangle is 34cm. find the length of the two legs, if one leg is 14cm longer than the other.
The rule of Pythagoras:
c ^ 2 = a ^ 12 + b ^ 2
In this case :
c = 34 cm
b = a + 14
c = a ^ 2 + ( a + 14 ) ^ 2
34 ^ 2 = a ^ 2 + ( a + 14 ) ^ 2
1156 = a ^ 2 + a ^ 2 + 2 * a * 14 + 14 ^ 2
1156 = 2 a ^ 2 + 28 a + 196 Subtract 196 to both sides
1156 - 196 = 2 a ^ 2 + 28 a + 196 - 196
960 = 2 a ^ 2 + 28 a Divide both sides by 2
480 = a ^ 2 + 14 a Add 49 to both sides
480 + 49 = a ^ 2 + 14 a + 49
529 = ( a + 7 ) ^ 2
________________________________________
Becouse :
( a + 7 ) ^ 2 = a ^ 2 + 2 * a * 7 + 7 ^ 2 =
a ^ 2 + 14 a + 49
________________________________________
sqrt ( 529 ) = sqrt [ ( a + 7 ) ^ 2 ]
+ OR - 23 = a + 7
1 case:
23 = a + 7 Subtract 7 to both sides
23 - 7 = a + 7 - 7
16 = a
a = 16 cm
2 case:
- 23 = a + 7 Subtract 7 to both sides
- 23 - 7 = a + 7 - 7
- 30 = a
a = - 30 cm
Length can't be negative so:
a = 16 cm
b = a + 14 = 16 + 14 = 30 cm
Well, well, well, we have ourselves a right triangle problem, don't we? 🤡
Let's say one leg of the right triangle is x cm. According to the problem, the other leg is 14 cm longer, which makes it (x + 14) cm.
Now, it's time to whip out the Pythagorean theorem! According to that theorem, the sum of the squares of the two legs is equal to the square of the hypotenuse.
So, we can write the equation as:
x^2 + (x + 14)^2 = 34^2
But wait, before we go any further, let me just grab my calculator. Ah, there it is! Now, let's do some number crunching.
Solving the equation will give us x ≈ 20.9 cm (approximately) for the shorter leg. And since the longer leg is 14 cm longer, it will be approximately x + 14 ≈ 34.9 cm.
So, the length of the shorter leg is approximately 20.9 cm and the length of the longer leg is approximately 34.9 cm.
Remember, these values are just approximations, so don't go trimming your legs to match them exactly! 😄
Let's assume the length of one of the legs of the right triangle is x cm.
According to the given information, the other leg will be x + 14 cm.
Now, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the two legs (a and b).
Using this formula, we can write the equation as follows:
x^2 + (x + 14)^2 = 34^2
Expanding and simplifying this equation:
x^2 + (x^2 + 28x + 196) = 1156
Combining like terms:
2x^2 + 28x + 196 = 1156
Subtracting 1156 from both sides:
2x^2 + 28x - 960 = 0
Now let's solve this quadratic equation using factoring or the quadratic formula.
Factoring the expression 2x^2 + 28x - 960:
2(x^2 + 14x - 480) = 0
Now let's find the two possible values of x:
Setting each factor equal to zero:
2(x + 40)(x - 12) = 0
This gives us two possible solutions: x + 40 = 0 and x - 12 = 0
Solving for each value of x:
x + 40 = 0 -> x = -40 cm (not a valid solution in this context)
x - 12 = 0 -> x = 12 cm
Therefore, one leg of the right triangle is 12 cm and the other leg is 12 + 14 = 26 cm.
To find the lengths of the two legs of a right triangle, we can use the Pythagorean theorem, which states that in a right 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 one leg of the triangle is x cm long. Since the other leg is 14 cm longer, its length will be (x + 14) cm.
According to the Pythagorean theorem, we have the equation:
x^2 + (x + 14)^2 = 34^2
Simplifying this equation, we get:
x^2 + (x^2 + 28x + 196) = 1156
Combining like terms:
2x^2 + 28x + 196 = 1156
Rearranging this equation, we have a quadratic equation:
2x^2 + 28x + 196 - 1156 = 0
2x^2 + 28x - 960 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula. Factoring it, we get:
2(x^2 + 14x - 480) = 0
(x + 30)(x - 16) = 0
So, either x + 30 = 0 or x - 16= 0
If x + 30 = 0, then x = -30 (which is not possible in this context since lengths cannot be negative).
If x - 16 = 0, then x = 16.
Therefore, one leg of the triangle is 16 cm long, and since the other leg is 14 cm longer, its length is (16 + 14) = 30 cm.