what is the radius of the circle circumscribing an isosceles right triangle having an area of 162 sq. cm?
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the diameter of the circle is the hypotenuse of the right triangle.
The diagonal of an isosceles right triangle of side a is 2√a
To find the radius of the circle circumscribing an isosceles right triangle, we can use the formula for the area of a triangle.
1. Let's start by finding the side length of the isosceles right triangle. Since the triangle is isosceles, two sides are equal. Let's call the length of each equal side "a" and the length of the hypotenuse "c".
2. Since the triangle is a right triangle, we can use the Pythagorean theorem to relate the side lengths. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, we have:
a^2 + a^2 = c^2
Simplifying this equation, we get:
2a^2 = c^2
Taking the square root of both sides, we have:
√(2a^2) = √(c^2)
√2 * a = c
3. Now, let's find the area of the triangle. The formula for the area of a triangle is 1/2 * base * height. In this case, the base and height are both equal to "a". So we have:
Area = 1/2 * a * a
Area = 1/2 * a^2
Given that the area is 162 sq. cm, we can set up the equation:
1/2 * a^2 = 162
Multiplying both sides by 2 to get rid of the fraction:
a^2 = 324
Taking the square root of both sides:
a = √(324)
a = 18 cm
4. Finally, let's find the radius of the circle circumscribing the triangle. Since the triangle is isosceles, the radius of the circumscribed circle is the distance from the center of the circle to one of the triangle's vertices. In this case, it is the distance from the center to the right angle vertex. This distance is equal to 1/2 times the length of the hypotenuse. So we have:
Radius = 1/2 * c
Radius = 1/2 * √2 * a
Radius = 1/2 * √2 * 18 cm
Radius = 9√2 cm
Therefore, the radius of the circle circumscribing the isosceles right triangle with an area of 162 sq. cm is 9√2 cm.
To find the radius of the circle circumscribing an isosceles right triangle, we need to make use of the properties of the circumcircle and the area of the triangle.
Step 1: Determine the dimensions of the triangle.
Let's assume that the two equal sides of the isosceles right triangle have a length of x units each. Since the triangle is right-angled, the length of the hypotenuse (which is also the diameter of the circumcircle) can be determined using the Pythagorean theorem:
(x^2) + (x^2) = (Hypotenuse^2)
2x^2 = Hypotenuse^2
Hypotenuse = sqrt(2x^2) = sqrt(2) * x
Step 2: Calculate the area of the triangle.
The area of an isosceles right triangle is given by (1/2) * base * height. In this case, since the triangle is isosceles, the base and height are equal. So we have:
Area = (1/2) * x * x = x^2
Step 3: Relate the area to the circumcircle.
The area of a triangle can be related to the radius of its circumcircle using the formula:
Area = (1/2) * radius^2
So, in this case, we have:
x^2 = (1/2) * radius^2
2x^2 = radius^2
Step 4: Calculate the radius of the circumcircle.
To find the radius, we need to substitute the area value given in the question. In this case, the area is 162 sq. cm.
2x^2 = radius^2
2 * 162 = radius^2
324 = radius^2
Taking the square root of both sides, we get:
radius = sqrt(324)
radius = 18 cm
Therefore, the radius of the circle circumscribing the isosceles right triangle with an area of 162 sq. cm is 18 cm.