The base of an isosceles triangle is 4x cm long.The altitude to the base is 6cm long.Find the length of a leg of the triangle.

The leg will be the hypotenuse of a right-angled triangle with base 2x and height 6

H^2 = 4x^2 + 36
H = √(4x^2 + 36)
= 2√(x^2 + 9)

To find the length of a leg of the isosceles triangle, we can use the Pythagorean theorem.

Let's assume the length of each leg of the triangle is y cm.

Using the Pythagorean theorem, we have:

y^2 = (4x/2)^2 + 6^2

Simplifying the equation, we have:

y^2 = (2x)^2 + 36
y^2 = 4x^2 + 36

To solve for y, we take the square root of both sides:

y = √(4x^2 + 36)

Therefore, the length of a leg of the isosceles triangle is √(4x^2 + 36) cm.

To find the length of a leg of the isosceles triangle, we can use the Pythagorean theorem.

The Pythagorean theorem states that in a right 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 other two sides.

In this case, we can consider the altitude (6 cm) as the height of the isosceles triangle. The two legs of the triangle will be congruent, so let's denote the length of each leg as 'x', since we need to find its value.

Now, since we have an altitude to the base (not the hypotenuse), we can draw a right triangle by considering one leg as the base, the altitude as the height, and the other leg as the hypotenuse.

So, using the Pythagorean theorem, we have:
x^2 + 6^2 = (4x)^2

Expanding this equation, we get:
x^2 + 36 = 16x^2

Moving all the terms to one side, we have:
16x^2 - x^2 - 36 = 0

Simplifying, we get:
15x^2 - 36 = 0

To solve for x, we can factor the quadratic equation:
(5x + 6)(3x - 6) = 0

Setting each factor equal to zero and solving for x, we find two possible solutions: x = -6/5 or x = 6/3. Since the length cannot be negative, we can ignore the first solution.

Thus, the length of each leg of the isosceles triangle is 6/3 cm, which simplifies to 2 cm.