LINGKARAN DALAM SEGITIGA DAN LINGKARAN LUAR SEGITIGA

Summary

This video explains the concepts of inscribed and circumscribed circles in a triangle. The inscribed circle touches all three sides of the triangle, while the circumscribed circle passes through all three vertices. The script covers how to calculate the radii for both circles using formulas involving the sides and area of the triangle. The example problems demonstrate how to apply these formulas to right triangles, including a step-by-step calculation of the radius of both types of circles. By the end, viewers will understand the relationship between a triangle's dimensions and its associated circles.

Takeaways

• 😀 A circle inscribed in a triangle touches all three sides of the triangle, and these sides are tangent to the circle.
• 😀 The radius of the inscribed circle (denoted as 'r') can be calculated using the formula: r = l / s, where 'l' is the area of the triangle and 's' is half the perimeter.
• 😀 A circle circumscribed around a triangle passes through all three vertices of the triangle.
• 😀 The radius of the circumscribed circle (denoted as 'R') is calculated using the formula: R = (a * b * c) / (4 * A), where 'a', 'b', and 'c' are the triangle's side lengths, and 'A' is the area of the triangle.
• 😀 In a right-angled triangle, the Pythagorean theorem can be used to find the missing side length when two sides are known.
• 😀 The area of a triangle can be found using Heron's formula, where the semi-perimeter 's' is half of the perimeter, and the area 'A' is calculated as A = √(s * (s - a) * (s - b) * (s - c)).
• 😀 For an example problem with a right-angled triangle, the radius of the inscribed circle is calculated using the formula r = l / s.
• 😀 For another example, the radius of the circumscribed circle is calculated using the formula R = (a * b * c) / (4 * A).
• 😀 The ratio of the radii of the circumscribed and inscribed circles is given by R / r, and can be computed once both radii are known.
• 😀 The solution to the example problems demonstrates how to apply these formulas step by step to calculate the radii of both the inscribed and circumscribed circles.

Q & A

What is the in-circle of a triangle?

The in-circle of a triangle is a circle that is inscribed within the triangle, where each side of the triangle is tangent to the circle. It touches all three sides of the triangle at one point each.

How do we calculate the radius of the in-circle?

The radius of the in-circle, denoted as r_d, can be calculated using the formula r_d = L / s, where L is the area of the triangle and s is the semi-perimeter (half of the perimeter).

What is the circumcircle of a triangle?

The circumcircle of a triangle is a circle that passes through all three vertices of the triangle. The center of this circle is the circumcenter, and the radius is called the circumradius.

How do we calculate the radius of the circumcircle?

The radius of the circumcircle, denoted as r_l, can be calculated using the formula r_l = (a * b * c) / (4 * L), where a, b, and c are the sides of the triangle and L is the area of the triangle.

What is the semi-perimeter of a triangle?

The semi-perimeter, denoted as s, is half of the perimeter of the triangle. It is calculated as s = 1/2 * (a + b + c), where a, b, and c are the sides of the triangle.

How do we calculate the area of a triangle using Heron's formula?

The area of a triangle can be calculated using Heron's formula: L = √[s * (s - a) * (s - b) * (s - c)], where a, b, and c are the sides of the triangle, and s is the semi-perimeter.

What is the relationship between the in-radius and circum-radius in a triangle?

The in-radius (r_d) and circum-radius (r_l) are two different radii of circles associated with the triangle. The ratio of the circumradius to the inradius (r_l / r_d) can provide insights into the shape and proportions of the triangle.

How do we determine the in-radius and circum-radius in a right-angled triangle?

In a right-angled triangle, the in-radius (r_d) can be found using the formula r_d = L / s, where L is the area and s is the semi-perimeter. The circum-radius (r_l) is given by r_l = (a * b * c) / (4 * L), where a, b, and c are the sides of the triangle.

In the example with triangle ABC, what is the in-radius for a right-angled triangle?

In the right-angled triangle ABC with sides AB = 20 cm, BC = 15 cm, and AC = 25 cm, the in-radius (r_d) was calculated to be 5 cm.

What is the significance of the Pythagorean theorem in the example provided?

The Pythagorean theorem is used to calculate the missing side of a right-angled triangle. In the example, it helped determine the length of side BC, which was needed to proceed with further calculations of the area and radii.