Problem 179
Question
\(C\) is right-angled and a perpendicular \(C D\) is drawn to \(A B\). The radii of the circles inscribed into the triangles \(A C D\) and \(B C D\) are equal to \(x\) and \(y\) respectively. Find the radius of the circle inscribed into the triangle \(A B C\).
Step-by-Step Solution
Verified Answer
The radius of the circle inscribed into the triangle ABC is \(r = \frac{2xy}{2x+2y}\).
1Step 1: Express the sides in terms of radii of inscribed circles
Let's denote the length of \(AB\) as \(a\), \(AC\) as \(b\), and \(BC\) as \(c\). We can express \(a\), \(b\), and \(c\) in terms of \(x\) and \(y\) because for a right-angled triangle the radius of the inscribed circle is half the sum of the two sides forming the right angle minus the hypotenuse. In this case, this gives us \(AC=2x\), \(BC=2y\), and \(AB=2x+2y\).
2Step 2: Find the semi-perimeter of triangle ABC
The semi-perimeter of a triangle is half of the sum of the lengths of its sides. For triangle ABC this is \(s = (a+b+c)/2 = (2x+2y+2x+2y)/2 = 2x+2y\).
3Step 3: Use Heron's formula to find the area of triangle ABC
Heron's formula gives the area of a triangle when the lengths of its sides are known. It is \(\sqrt{s(s-a)(s-b)(s-c)}\). Substituting the values we have for \(a\), \(b\), \(c\), and \(s\), this gives us \(\sqrt{(2x+2y)(2y)(2x)(2x+2y)} = 2xy\).
4Step 4: Calculate the radius of the inscribed circle
The radius of an inscribed circle in a right-angle triangle can be calculated as \(r = \frac{Area of triangle}{Semiperimeter of triangle}\). Substituting our expressions for for the area of triangle ABC and the semi-perimeter, this gives us \(r = \frac{2xy}{2x+2y}\).
Key Concepts
Heron's FormulaSemi-Perimeter of a TriangleRight-Angled Triangle Properties
Heron's Formula
Understanding Heron's formula is crucial for calculating the area of a triangle when you only know the lengths of its sides. Simply put, it is a mathematical way to find the area without knowing the height. The formula is expressed as \( A = \sqrt{s(s - a)(s - b)(s - c)} \), where \( A \) is the area, \( a, b, \) and \( c \) are the lengths of the sides, and \( s \) is the semi-perimeter of the triangle.
Heron's formula is especially helpful in situations where the lengths of all sides of the triangle are known, but the altitude is not, as is often the case in geometric problems. Using this formula requires calculating the semi-perimeter first, which will be further explained in the next section.
Heron's formula is especially helpful in situations where the lengths of all sides of the triangle are known, but the altitude is not, as is often the case in geometric problems. Using this formula requires calculating the semi-perimeter first, which will be further explained in the next section.
Semi-Perimeter of a Triangle
The semi-perimeter of a triangle is essentially half the perimeter, which is why you might sometimes see it referred to as the 'half-perimeter'. To calculate it, you sum up the lengths of all three sides of the triangle and divide by two, expressed as \( s = \frac{a + b + c}{2} \), where \( a, b, \) and \( c \) represent the lengths of the triangle's sides.
Why is the Semi-Perimeter Important?
Calculating the semi-perimeter is a critical step in many geometric formulas, including Heron's formula for the area of a triangle. It's a strategic intermediary value which simplifies computations when dealing with triangles. This concept is particularly helpful for problems that involve the triangle's area, such as finding the radius of the inscribed circle.Right-Angled Triangle Properties
Triangles with a right angle, known as right-angled triangles, have unique properties that are widely used in geometry. The right angle specifically impacts key computations, such as the relationship between the triangle's sides (Pythagorean theorem) and the radius of the inscribed circle.
The radius of an inscribed circle in a right-angled triangle is given by \( r = \frac{a + b - c}{2} \), where \( a \) and \( b \) are the lengths of the two legs forming the right angle, and \( c \) is the length of the hypotenuse. This formula demonstrates a distinctive property: the radius of the inscribed circle relies on the lengths of the sides but is calculated in a different manner than for other triangles.
Moreover, the relationship between the sides of a right-angled triangle, defined by the Pythagorean theorem \( a^2 + b^2 = c^2 \), ensures the sides maintain a consistent relation, which further influences the geometric characteristics of the triangle, including the size of the inscribed circle.
The radius of an inscribed circle in a right-angled triangle is given by \( r = \frac{a + b - c}{2} \), where \( a \) and \( b \) are the lengths of the two legs forming the right angle, and \( c \) is the length of the hypotenuse. This formula demonstrates a distinctive property: the radius of the inscribed circle relies on the lengths of the sides but is calculated in a different manner than for other triangles.
Moreover, the relationship between the sides of a right-angled triangle, defined by the Pythagorean theorem \( a^2 + b^2 = c^2 \), ensures the sides maintain a consistent relation, which further influences the geometric characteristics of the triangle, including the size of the inscribed circle.
Other exercises in this chapter
Problem 175
Let \(A B C\) be a triangle having \(O\) and \(I\) as its circumcentre and incentre respectively. If \(R\) and \(r\) are the circumradius and the inradius respe
View solution Problem 178
Prove that the area of the incircle is to the area of the triangle itself is \(\pi: \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2}\).
View solution Problem 180
If a circle be drawn touching the incircle and circumcircle of a triangle and the side \(B C\) externally, prove that it's radius is \(\frac{\Delta}{a} \tan ^{2
View solution Problem 181
\(\frac{r r_{1}}{r_{2} r_{3}}=\tan ^{2} \frac{A}{2}\)
View solution