The hypotenuse of a right angled triangle is 20 m. If the difference between the length of the other sides be 4m. find the other sides?
This is a right triangle problem with
c^2 = a^2 + b^2
The difference between a and b is 4.
a- b = 4 or a = b +4
20^2 = (b+4)^2 + b^2
To find the answer you have to multiply (b+4)(b+4) and add b^2.
Next subtract the 400 from both sides so that you have 0 = on one side.
Factor and solve for b. Then you can find a.
The sides of a hypotenuse are all multiples of the original Pythagorean triple: 3, 4, 5. Obviously in this, 5 is the hypotenuse and in your equation so is 20. To get 20, you have to multiply 5 by four so to get the Pythagorean triple you just need to multiply the original set by 4. So your answer will be 12, 16 and 20.
Let's denote the length of the two other sides of the right-angled triangle as x and y.
According to the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides, we can write the following equation:
x^2 + y^2 = 20^2
Given that the difference between the length of the other sides is 4m, we can write another equation:
x - y = 4
Now, we have a system of two equations with two variables. To solve this system, we can use substitution or elimination. Let's use substitution:
From the second equation, we can express x in terms of y:
x = y + 4
Substituting this into the first equation, we get:
(y + 4)^2 + y^2 = 20^2
Expanding and simplifying:
y^2 + 8y + 16 + y^2 = 400
2y^2 + 8y + 16 = 400
Dividing through by 2:
y^2 + 4y + 8 = 200
Rearranging and simplifying:
y^2 + 4y - 192 = 0
Now, we can solve this quadratic equation for y. Factoring the equation:
(y + 16)(y - 12) = 0
Setting each factor to zero and solving for y:
y + 16 = 0 --> y = -16
y - 12 = 0 --> y = 12
Since lengths cannot be negative, we can discard the solution y = -16.
Therefore, the length of the other side is y = 12 m.
Substituting this value back into x = y + 4, we find:
x = 12 + 4 = 16 m
So, the lengths of the other sides of the right-angled triangle are 12 m and 16 m.
To find the lengths of the other sides of a right-angled triangle given the hypotenuse and the difference between the lengths of the other sides, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a 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. This can be expressed as:
c^2 = a^2 + b^2
where c is the hypotenuse, and a and b are the other sides.
In this case, we are given that the hypotenuse, c, is 20 m. Let's use this information to set up the equation.
20^2 = a^2 + b^2
Simplifying the equation, we get:
400 = a^2 + b^2
We are also given that the difference between the lengths of the other two sides is 4 m. Mathematically, we can express this as:
a - b = 4
Now we have a system of two equations with two variables (a and b). We can solve this system of equations simultaneously to find their values.
Let's rearrange the second equation to express one variable in terms of the other:
a = b + 4
Now substitute this expression for a in the first equation:
400 = (b + 4)^2 + b^2
Expanding and simplifying the equation, we get:
400 = b^2 + 8b + 16 + b^2
Combining like terms:
2b^2 + 8b + 16 - 400 = 0
Simplifying further:
2b^2 + 8b - 384 = 0
Dividing through by 2 to simplify the equation:
b^2 + 4b - 192 = 0
Now we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, let's use factoring.
Factoring the equation, we find:
(b + 16)(b - 12) = 0
This equation is satisfied when either (b + 16) = 0 or (b - 12) = 0:
b + 16 = 0 or b - 12 = 0
Solving for b, we get:
b = -16 or b = 12
Since the length of a side cannot be negative, we discard the solution b = -16.
Therefore, b = 12 m.
Substituting this value for b back into the equation a = b + 4, we can find a:
a = 12 + 4
a = 16 m
So, the lengths of the other sides of the right-angled triangle are 16 m and 12 m.