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anemone
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Suppose the lengths of the three sides of $\triangle ABC$ are integers and the inradius of the triangle is 1. Prove that the triangle is a right triangle.
A right triangle is a type of triangle that has one angle measuring 90 degrees. It is also known as a right-angled triangle.
To prove that a triangle ABC is a right triangle, you can use the Pythagorean Theorem, which 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 other two sides. If the equation a² + b² = c² holds true, then the triangle is a right triangle.
Apart from the Pythagorean Theorem, you can also use the converse of the Pythagorean Theorem, which states that if the square of the length of the longest side is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. Additionally, you can use trigonometric ratios, such as sine, cosine, and tangent, to determine if the triangle is a right triangle.
Yes, a triangle can still be a right triangle even if it has two equal sides. This type of triangle is called an isosceles right triangle. In this case, the two equal sides are the legs of the triangle, and the remaining side is the hypotenuse.
Proving that a triangle is a right triangle is important in geometry because it allows you to accurately determine the measurements and properties of the triangle. It also helps in solving real-world problems, such as calculating distances and heights using trigonometric functions.