In a right angled triangle, the length of a hypotenuse is 10 cm. Of the two shorter sides, one side is 2 cm longer than the other side. i) Find the lengths of the two shorter sides. ii)Hence, find the area of the triangle.
this is a Pythagorean triple
... 3-4-5 times 2 ... 6-8-10
x^2 + (x + 2)^2 = 10^2
2x^2 + 4x - 96 = 0
x^2 + 2x - 48 = 0
(x + 8) (x - 6) = 0
x = 6 ... x + 2 = 8
1m
To find the lengths of the two shorter sides of a right-angled triangle, we can use the Pythagorean theorem, which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's call the lengths of the two shorter sides "x" and "x + 2" cm.
According to the Pythagorean theorem, we have:
x^2 + (x + 2)^2 = 10^2
Now we can solve this equation to find the values of "x" and "x + 2".
Expanding the equation, we get:
x^2 + x^2 + 4x + 4 = 100
Combining like terms, we have:
2x^2 + 4x + 4 = 100
Subtracting 100 from both sides, we get:
2x^2 + 4x - 96 = 0
Dividing both sides by 2, we get:
x^2 + 2x - 48 = 0
Now we can factorize the quadratic equation:
(x + 8)(x - 6) = 0
Setting each factor equal to zero, we have two possible solutions:
x + 8 = 0 or x - 6 = 0
Solving for x in each case, we get:
x = -8 or x = 6
Since the length of a side cannot be negative, we can ignore the first solution. Therefore, x = 6.
Hence, the lengths of the two shorter sides are 6 cm and 8 cm (x = 6, x + 2 = 8).
Now, to find the area of the triangle, we can use the formula:
Area = (1/2) * base * height
In this case, the base is one of the shorter sides (6 cm), and the height is the other shorter side (8 cm).
Therefore, the area of the triangle is:
Area = (1/2) * 6 cm * 8 cm = 24 cm^2
So, the area of the triangle is 24 square cm.