answer:The given triangle is a classic example of a Pythagorean triple. Given the side lengths, we can use the properties of right-angled triangles to find the circumradius and inradius. First, let's establish that the given triangle is indeed a right-angled triangle. We can verify this using the Pythagorean theorem, which states that for a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, we have a^2 + b^2 = 3^2 + 4^2 = 25, which is equal to c^2 = 5^2. This confirms that the triangle is right-angled. Now, let's find the circumradius. The circumradius of a right-angled triangle is half the length of the hypotenuse. In this case, the circumradius is R = c/2 = 5/2. To find the inradius, we can use the formula r = (a + b - c)/2, where r is the inradius, or we can use the area formula to derive the inradius. The area of a triangle can be expressed as A = rs, where r is the inradius and s is the semiperimeter. The semiperimeter is given by s = (a + b + c)/2 = (3 + 4 + 5)/2 = 6. The area of a right-angled triangle can also be found using the formula A = (ab)/2. Equating these two expressions for the area, we get (ab)/2 = rs, which gives r = ab/(2s). Substituting the values, we get r = (3)(4)/(2)(6) = 2/2 = 1. Therefore, the circumradius of the triangle is 5/2, and the inradius is 1.

answer:Using the formulas R = abc / (4 * area) to find the circumradius and then A = rs to find the inradius is a more elegant and unified approach. We can calculate the area of the triangle using the formula A = (ab)/2, which gives A = (3)(4)/2 = 6. Then, using R = abc / (4 * area), we can find the circumradius: R = (3)(4)(5) / (4 * 6) = 60 / 24 = 5/2. Next, we can use the relation A = rs, where A is the area, r is the inradius, and s is the semiperimeter. The semiperimeter is given by s = (a + b + c)/2 = 6. Substituting the values, we get 6 = r(6), which gives r = 1. Alternatively, we can also use the relation r = (a + b - c)/2 or r = A / s, but using R = abc / (4 * area) and A = rs provides a nice connection between the circumradius and inradius, and allows us to derive both quantities in a more cohesive manner.

answer:The formula that connects the inradius r and the circumradius R of a triangle is given by: rR = (abc) / (4s) or, using another representation of this relationship, rR = Δ / s where a, b, and c are the side lengths of the triangle, s is the semiperimeter, and Δ is the area of the triangle.

answer:The Pythagorean Theorem is a fundamental concept in Euclidean geometry, describing the relationship between the lengths of the sides of a right-angled triangle. The theorem states that the square of the length of the hypotenuse, the side opposite the right angle, is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. This theorem is a cornerstone of geometry and trigonometry, and its applications extend far beyond the realm of mathematics, influencing various fields such as physics, engineering, and architecture. The theorem's significance lies in its ability to provide a straightforward method for calculating distances and lengths in right-angled triangles, making it an essential tool for solving problems in spatial reasoning and quantitative analysis. For instance, if you know the lengths of the two shorter sides of a right-angled triangle, the Pythagorean Theorem enables you to easily calculate the length of the hypotenuse, which is crucial in various real-world applications. One of the fascinating aspects of the Pythagorean Theorem is its numerous proofs, with over 350 known demonstrations of the theorem's validity. These proofs range from simple geometric constructions to more complex algebraic manipulations, illustrating the theorem's profound connections to various areas of mathematics. Moreover, the Pythagorean Theorem is not limited to Euclidean geometry; it has analogs in non-Euclidean geometries, such as the hyperbolic and elliptic planes. This highlights the theorem's far-reaching implications and the intrinsic beauty of mathematical concepts that transcend specific geometrical frameworks. In summary, the Pythagorean Theorem is a fundamental concept in geometry that has far-reaching implications in various fields of mathematics and beyond. Its simplicity, elegance, and versatility have made it a cornerstone of mathematical reasoning, and its numerous proofs continue to inspire mathematicians and scholars to explore the intricacies of geometric relationships.