The perimeter of an isosceles triangle is 42 cm . its base is 2/3 times the sum of the equal sides . find the length of each side and the area of the triangle.

let each of the equal sides be x

let the base by y
y = (2/3)(2x) = 4x/3

so x + x + (4/3)x = 42
multipy by 3
3x + 3x + 4x = 126
10x = 126
x = 12.6
so the base = (4/3)(12.6) = 16.8

use Pythagoras to find the height, h
h^2 + 8.4^2 = 12.6^
h^2 = 88.2
h √88.2

Area = (1/2)base x height
= (1/2)(16.8)√88.2
= appr. 78.888

the perimeter of an isosceles triangle is 42cm and its bases is(3/2) times each of the equal sides .find the length of each side of the triangle ;area of the triangle and the height of the triangle.

Let's assume the length of each equal side of the isosceles triangle is 'x'. Therefore, the base of the triangle would be (2/3)(x + x) = (2/3)(2x) = (4/3)x.

The perimeter of the triangle is the sum of all three sides. Therefore,
2x + (4/3)x + (4/3)x = 42 cm.

To simplify the equation, we can multiply through by 3:
6x + 4x + 4x = 126.

Combine like terms:
14x = 126.

Divide both sides by 14:
x = 9.

So, the length of each equal side is 9 cm, and the base length is (4/3)(9) = 12 cm.

To find the area of the triangle, we can use the formula A = (1/2)bh, where b is the base length and h is the height. Since this is an isosceles triangle, the height can be found using the Pythagorean theorem.

Let's draw a line from the top vertex to the base, creating two right triangles.
Each right triangle will have a base length of 6 cm (half of the base of 12 cm) and hypotenuse length of 9 cm (one of the equal sides).

To find the height (h) of each right triangle, we can use the Pythagorean theorem:
h^2 = 9^2 - 6^2
h^2 = 81 - 36
h^2 = 45
h = √45
h = 3√5.

Since the height of each right triangle is also the height of the isosceles triangle, the area can be calculated as follows:
A = (1/2)(12)(3√5)
A = 6√5 cm^2.

Therefore, the length of each side of the isosceles triangle is 9 cm and the area of the triangle is 6√5 cm^2.

To find the length of each side and the area of the isosceles triangle, we will follow these steps:

Step 1: Let's assume the length of each equal side of the triangle is "x" cm.
Step 2: Since the base is 2/3 times the sum of the equal sides, the length of the base is (2/3)(2x) = (4/3)x cm.
Step 3: The perimeter of a triangle is the sum of all its sides. In this case, the perimeter is the sum of the length of the equal sides and the length of the base, which is x + x + (4/3)x = 42 cm.
Step 4: Simplifying the equation, we get (2 + 4/3)x = 42 cm.
Step 5: Multiplying 4/3 by 2, we get (8/3)x = 42 cm.
Step 6: To isolate x, we divide both sides of the equation by 8/3:

(8/3)x / (8/3) = 42 cm / (8/3),
x = (42 cm * 3) / 8,
x = 15.75 cm.

Therefore, the length of each equal side is 15.75 cm.

Step 7: Now we can find the length of the base by substituting x into the expression we obtained earlier:
Base = (4/3)x = (4/3)(15.75 cm) = 21 cm.

So, the length of the base is 21 cm.

Step 8: To find the area of the triangle, we can use the formula for the area of an isosceles triangle:

Area = (1/2) * base * height.

Since the triangle is isosceles, the height will be the perpendicular distance from the top vertex to the base. We need to find this height.

By using the Pythagorean theorem, we can determine that the height is:

height = sqrt((15.75 cm)^2 - (10.5 cm)^2),
height = sqrt(246.75 cm^2 - 110.25 cm^2),
height = sqrt(136.5 cm^2),
height ≈ 11.67 cm.

Now we can calculate the area of the triangle:

Area = (1/2) * 21 cm * 11.67 cm,
Area ≈ 244.4175 cm^2.

So, the area of the triangle is approximately 244.4175 cm^2.