A piece of flat pastry is cut in the shape of a right-angled triangle. The longest side is 6b cm and the
shortest is 2b cm
a) Find the length of the third side.
b) Prove that the area of the triangle is equal to 4√2b^2 cm^2.
a) To find the length of the third side of the right-angled triangle, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the longest side is 6b cm and the shortest side is 2b cm.
Let the length of the third side be c cm.
According to the Pythagorean theorem, we have:
(6b)^2 = (2b)^2 + c^2
Simplifying this equation, we get:
36b^2 = 4b^2 + c^2
Combining like terms, we have:
32b^2 = c^2
Taking the square root of both sides, we get:
c = √(32b^2)
Simplifying further, we have:
c = 4√2b cm
Therefore, the length of the third side of the triangle is 4√2b cm.
b) To prove that the area of the triangle is equal to 4√2b^2 cm^2, we need to use the formula for the area of a right-angled triangle.
The area of a triangle is given by the formula:
Area = (1/2) * base * height
In our case, the third side, which we found to be 4√2b cm, can be considered as the base of the triangle.
Let the height of the triangle be h cm.
The formula for the area becomes:
Area = (1/2) * (4√2b) * h
Simplifying this equation, we have:
Area = 2√2b * h
To prove that the area is equal to 4√2b^2 cm^2, we need to find the value of h in terms of b.
Since the shortest side of the triangle is 2b cm, we can consider it as the height of the triangle.
Therefore, h = 2b cm.
Substituting this value of h into our equation for the area, we get:
Area = 2√2b * 2b
Simplifying further, we have:
Area = 4√2b^2 cm^2
Hence, we have proved that the area of the triangle is equal to 4√2b^2 cm^2.
To find the length of the third side of the right-angled triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.
In this case, let's denote the length of the third side as c cm. According to the given information, the longest side is 6b cm (hypotenuse) and the shortest side is 2b cm. Applying the Pythagorean theorem, we have:
c^2 = (6b)^2 - (2b)^2
c^2 = 36b^2 - 4b^2
c^2 = 32b^2
Taking the square root of both sides, we get:
c = √(32b^2)
Simplifying further:
c = √(4 * 8 * b^2)
c = 2√8b
c = 2√(4 * 2 * b)
c = 2 * 2√2b
c = 4√2b
Therefore, the length of the third side is 4√2b cm.
To prove that the area of the triangle is equal to 4√2b^2 cm^2, we can use the formula for the area of a right-angled triangle:
Area = 0.5 * base * height
In this case, the base is 2b and the height is 6b. Plugging these values into the formula, we get:
Area = 0.5 * (2b) * (6b)
Area = b * 3b
Area = 3b^2
Now, let's calculate the given expression for the area:
4√2b^2 = 4 * √(2b^2)
= 4 * √(2 * b * b)
= 4 * √2 * √(b * b)
= 4 * √2 * b
We can simplify 4 * √2 to approximate value 5.66. Now, let's compare the simplified given expression with the calculated area:
4 * √2 * b ≈ 5.66 * b
We can see that 5.66 * b is approximately equal to 3b, which is the calculated area of the triangle. Therefore, we have proven that the area of the triangle is equal to 4√2b^2 cm^2.
x^2 + (2b)^2 = (6b)^2
x^2 + 4b^2 = 36b^2
x^2 = 32b^2
x = 4b√2
Now recall that area = bh/2 and just do the multiplication