if right angle haveing perimeter=36,diagnol=15 then area=?
To find the area of a right angle triangle given its perimeter and diagonal, we need to solve the problem using geometry and algebra.
Let's start by labeling the triangle. Let's call the two legs of the triangle a and b, and the hypotenuse c. The perimeter of the triangle is the sum of the lengths of all three sides, so we have:
Perimeter = a + b + c
In this case, the perimeter is given as 36, so we can write the equation as:
36 = a + b + c .......(Equation 1)
The diagonal of a right-angled triangle divides it into two congruent right-angled triangles. Therefore, we can use the Pythagorean theorem to relate the sides of these two smaller triangles.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, since the hypotenuse c is the diagonal, we have:
c^2 = a^2 + b^2
Substituting the values given in the problem, we have:
15^2 = a^2 + b^2
225 = a^2 + b^2 .......(Equation 2)
Now we have two equations, Equation 1 and Equation 2, which we can solve simultaneously to find the values of a and b. Once we know the values of a and b, we can calculate the area of the right-angled triangle.
There are several methods to solve a system of equations, including substitution, elimination, or using matrices. In this case, substitution is the simplest approach.
From Equation 1, we have:
a + b = 36 - c
Substituting this value in Equation 2, we get:
225 = (36 - c)^2 + b^2
Expanding the equation, we have:
225 = 1296 - 72c + c^2 + b^2
Rearranging the terms, we have:
b^2 = 225 - 1296 + 72c - c^2
Simplifying, we get:
b^2 = -1071 + 72c - c^2
Now, we have a quadratic equation in terms of b^2. To solve for b, we need to determine the value of c by using Equation 1:
c = 36 - a - b
Substituting this value in the quadratic equation for b^2, we get:
b^2 = -1071 + 72(36 - a - b) - (36 - a - b)^2
Expanding and simplifying the equation, we get:
b^2 = -1071 + 2592 - 72a - 72b - 36^2 + 2(36)(a) + 2(36)(b) - a^2 - 2(a)(b) - b^2
Rearranging the terms and simplifying, we have:
2b^2 + 2ab - 108a - 72b + 360 - a^2 = 0
Now we have a quadratic equation in terms of a and b. We can solve this equation using various methods, such as factoring, completing the square, or using the quadratic formula.
Once we find the values of a and b, we can calculate the area of the right-angled triangle using the formula:
Area = (1/2) * base * height
In a right-angled triangle, the two legs can be considered as the base and height, so we have:
Area = (1/2) * a * b
Substituting the values of a and b, we can finally find the area of the right-angled triangle.