the hypotenuse of a right angled triangle is 20cm. the difference between its other two sides be 4cm. The sides are
16cm and 14cm
x^2 + (x+4)^2 = 20^2
Think of a scaled-up 3-4-5 triangle.
Let's assume the lengths of the two sides adjacent to the right angle are x and x + 4. According to the Pythagorean theorem, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
Using this information, we can form the equation:
x^2 + (x + 4)^2 = 20^2
Expanding the equation:
x^2 + (x^2 + 8x + 16) = 400
Combining like terms:
2x^2 + 8x + 16 = 400
Subtracting 400 from both sides:
2x^2 + 8x - 384 = 0
Dividing the equation by 2 to simplify it:
x^2 + 4x - 192 = 0
Factoring the equation:
(x + 16)(x - 12) = 0
Setting each factor equal to zero:
x + 16 = 0 or x - 12 = 0
Solving for x:
x = -16 or x = 12
Since the length of a side cannot be negative, x = 12.
Therefore, the lengths of the sides of the right-angled triangle are 12cm, and 12 + 4 = 16cm.
To find the lengths of the other two sides of the right-angled triangle, you can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse 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 a and b. According to the problem description, the hypotenuse (c) is 20 cm, and the difference between the other two sides (a and b) is 4 cm.
Using the Pythagorean theorem, we have the following equation:
c^2 = a^2 + b^2
Substituting the known values, we get:
(20)^2 = (a)^2 + (b)^2
400 = a^2 + b^2
Now, we are given that the difference between a and b is 4 cm. We can express this as:
a - b = 4
From this, we can derive another equation by squaring both sides:
(a - b)^2 = 4^2
Expanding this equation, we get:
a^2 - 2ab + b^2 = 16
Now, we have a system of two equations:
a^2 + b^2 = 400 (Equation 1)
a^2 - 2ab + b^2 = 16 (Equation 2)
To solve this system of equations, we can substitute the value of (a^2 + b^2) in Equation 2 from Equation 1:
400 - 2ab = 16
Rearranging this equation to solve for ab, we get:
2ab = 400 - 16
2ab = 384
ab = 192
Now that we have the value of ab, we can represent a in terms of b by rearranging the equation a - b = 4:
a = b + 4
Substituting this value of a in terms of b in the equation ab = 192, we get:
(b + 4)b = 192
b^2 + 4b - 192 = 0
Now, we have a quadratic equation. We can solve it by factoring or using the quadratic formula.
Factoring, we can write the equation as:
(b - 12)(b + 16) = 0
From this, we get two possible values for b:
b = 12 or b = -16
Since we are dealing with lengths, the value of b cannot be negative. Therefore, the only valid solution is b = 12 cm.
Now, we can substitute this value of b back into the equation a = b + 4 to find the value of a:
a = 12 + 4
a = 16 cm
Therefore, the lengths of the other two sides of the right-angled triangle are 16 cm and 12 cm.