a rectangle with side of 6 cm and 5 cm has the same area as an isoceles triangle with base length 12 cm.find the altitude to the base of the triangle and side of the triangle
the height of two tower are 40 feet and 10 feet.if they are 16 feet apart find the distance between their tops
for the triangle,
6*5 = 1/2 * 12 * h
Now, knowing h, the side length s can be found using
s^2 = 6^2 + h^2
For the towers, the distance z is
z^2 = 16^2 + (40-10)^2
To find the altitude and side of the triangle, we need to use the formula for the area of a triangle.
Given that the rectangle has sides of 6 cm and 5 cm, the area of the rectangle would be the product of its sides: Area_rectangle = length * width = 6 cm * 5 cm = 30 cm².
Now, let's find the area of the isosceles triangle. We know that the base length is 12 cm. Let's denote the altitude as 'h' and the side length as 's'. The area of a triangle is given by the formula: Area_triangle = 0.5 * base * height.
Since the area of the rectangle and the triangle are equal, we have the equation:
30 cm² = 0.5 * 12 cm * h
Let's solve for the altitude, h:
30 cm² = 6 cm * h
Dividing both sides by 6:
5 cm² = h
So, the altitude of the isosceles triangle is 5 cm.
Now, let's find the side length, s. Since the triangle is isosceles, it has two equal sides, which we'll call 's'. We can use the Pythagorean theorem to find the side length, s. The theorem states that in a right triangle, the square of the hypotenuse (the longest side, which is 's' in this case) is equal to the sum of the squares of the other two sides.
The two sides are the altitude, h, and half the base, which is 12 cm / 2 = 6 cm. Applying the Pythagorean theorem:
s² = h² + (0.5 * base)²
s² = (5 cm)² + (6 cm)²
s² = 25 cm² + 36 cm²
s² = 61 cm²
Taking the square root to both sides:
s ≈ √(61 cm²)
So, the side length (approximate value) of the isosceles triangle is √61 cm, and the altitude to the base is 5 cm.