a triangle has sides equal to 68 cm, 77 cm and 75 cm, respectively. find the area of the escribed circle tangent to the shortest side of the triangle.
this url shows how to figure the radius of an escribed circle.
https://math.stackexchange.com/questions/1674656/radius-of-an-e-circle-in-terms-of-triangles-sides-and-area
To find the area of the escribed circle tangent to the shortest side of the triangle, you can use the formula for the area of an escribed circle combined with the formula for the area of a triangle.
Step 1: Find the semi-perimeter of the triangle.
The semi-perimeter (s) of a triangle is calculated by adding up all the sides of the triangle and dividing by 2.
s = (68 + 77 + 75) / 2
s = 210 / 2
s = 105 cm
Step 2: Find the area of the triangle.
Using Heron's formula, the area (A) of a triangle with sides a, b, and c and semi-perimeter s can be calculated as:
A = sqrt(s * (s - a) * (s - b) * (s - c))
A = sqrt(105 * (105 - 68) * (105 - 77) * (105 - 75))
A = sqrt(105 * 37 * 28 * 30)
A ≈ sqrt(9774000)
A ≈ 3124.946 cm² (rounded to three decimal places)
Step 3: Find the radius of the escribed circle.
The radius of the escribed circle (r) can be found by dividing the area of the triangle by the semi-perimeter.
r = A / s
r ≈ 3124.946 / 105
r ≈ 29.762 cm (rounded to three decimal places)
Step 4: Calculate the area of the escribed circle.
The area (Ae) of the escribed circle can be calculated using the formula Ae = π * r².
Ae = π * (29.762)²
Ae ≈ π * 887.124
Ae ≈ 2794.464 cm² (rounded to three decimal places)
Therefore, the area of the escribed circle tangent to the shortest side of the triangle is approximately 2794.464 cm².
To find the area of the escribed circle tangent to the shortest side of the triangle, we can follow these steps:
Step 1: Find the semiperimeter of the triangle.
The semiperimeter of a triangle is calculated by adding up all the sides and dividing the sum by 2. In this case:
Semiperimeter = (68 cm + 77 cm + 75 cm) / 2 = 220 cm
Step 2: Find the length of the tangent segment.
The shortest side of the triangle is the side with length 68 cm. To find the length of the tangent segment, we can use the formula:
Tangent segment = (Triangle semiperimeter - Side length) / 2
Tangent segment = (220 cm - 68 cm) / 2 = 76 cm
Step 3: Find the area of the escribed circle.
The radius of the escribed circle is equal to the length of the tangent segment. So, the radius of the escribed circle is 76 cm. The formula to calculate the area of a circle is:
Area = π * (radius)^2
Therefore, the area of the escribed circle tangent to the shortest side of the triangle is:
Area = π * (76 cm)^2
You can calculate the final value using a calculator.