find the radius of the circle circumscribing an isosceles right triangle having an area of 162 sq. cm
well, r = s/√3
Why did the circle go to the doctor? Because it wasn't feeling "circumspicious"!
To find the radius of the circle circumscribing an isosceles right triangle, we can use the Pythagorean theorem and the fact that the radius of the circle is equal to half the length of the hypothenuse of the triangle.
Let's denote the length of the legs of the triangle as "a" and the length of the hypothenuse as "c". Since it is an isosceles right triangle, the length of each leg is the same.
According to the Pythagorean theorem, we have the equation:
a^2 + a^2 = c^2
Simplifying this equation, we get:
2a^2 = c^2
Since the triangle has an area of 162 sq. cm, we have:
(1/2) * a * a = 162
Simplifying this equation, we get:
a^2 = 324
Taking the square root of both sides, we find:
a = 18 cm
Substituting this value back into the equation 2a^2 = c^2, we find:
2 * (18^2) = c^2
Simplifying this equation, we get:
648 = c^2
Taking the square root of both sides, we find:
c = 18√2 cm
Since the radius of the circle is equal to half the length of the hypothenuse, we have:
radius = (1/2) * 18√2 cm
Simplifying this expression, we get:
radius = 9√2 cm
Therefore, the radius of the circle circumscribing the isosceles right triangle is 9√2 cm.
To find the radius of the circle circumscribing an isosceles right triangle, we need to know some properties of this type of triangle.
In an isosceles right triangle, the two legs are congruent and the hypotenuse is the side opposite the right angle. Let's assume the length of each leg is 'a', and the hypotenuse is 'c'.
The area of an isosceles right triangle can be calculated using the formula:
Area = (1/2) * base * height.
In this case, the base and height of the triangle are both equal to 'a'. Therefore, we can write:
162 = (1/2) * a * a.
Simplifying this equation, we get:
162 = (1/2) * a^2
324 = a^2.
To find the value of 'a', we take the square root of both sides:
sqrt(324) = sqrt(a^2)
18 = a.
Now that we have the length of one leg of the triangle, we can find the hypotenuse using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Using this theorem, we can write:
c^2 = a^2 + a^2
c^2 = 2a^2
c = sqrt(2) * a.
Since we already know the value of 'a' from the previous calculation, we can substitute it in:
c = sqrt(2) * 18
c ≈ 25.4558.
Finally, to find the radius of the circle circumscribing the triangle, we divide the hypotenuse by 2, since the radius is always half the length of the diameter:
radius = c/2
radius ≈ 25.4558/2
radius ≈ 12.7279 cm.
Therefore, the radius of the circle circumscribing the isosceles right triangle with an area of 162 sq. cm is approximately 12.7279 cm.