Solution of Triangles: Distance of circumcentre from the sides
Solution of Triangles: Distance of circumcentre from the sides
The circumcentre of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. It is also the center of the circumcircle, the circle that passes through all three vertices of the triangle. The distance of the circumcentre from the sides of the triangle is an important concept in triangle geometry and can be calculated using various formulas depending on the information available.
Properties of the Circumcentre
Before we delve into the distances, let's understand some key properties of the circumcentre:
- It is equidistant from all three vertices of the triangle.
- In an acute-angled triangle, the circumcentre lies inside the triangle.
- In a right-angled triangle, the circumcentre is at the midpoint of the hypotenuse.
- In an obtuse-angled triangle, the circumcentre lies outside the triangle.
Distance of Circumcentre from the Sides
The distance of the circumcentre (denoted as O) from the sides of the triangle can be determined using the formula for the distance from a point to a line in the coordinate plane. If the equation of the side is given by Ax + By + C = 0, and the coordinates of the circumcentre are (x_0, y_0), the distance d from the circumcentre to the side is given by:
[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} ]
However, in the context of triangle geometry, we often use the sides' lengths and other triangle properties to find these distances.
Formulas Involving Triangle Sides and Angles
Let's denote the sides of the triangle as a, b, and c, and the angles opposite to these sides as A, B, and C respectively. The circumradius (radius of the circumcircle) is denoted as R.
The distances from the circumcentre to the sides of the triangle can be calculated using the following formulas:
| Side | Distance from Circumcentre (d) |
Formula |
|---|---|---|
a |
d_a |
( d_a = R \cos A ) |
b |
d_b |
( d_b = R \cos B ) |
c |
d_c |
( d_c = R \cos C ) |
These formulas are derived from the fact that the circumcentre forms two congruent right-angled triangles with each side of the triangle.
Example
Consider a triangle ABC with sides a = 8, b = 6, and c = 7. Let's calculate the circumradius R and then find the distance from the circumcentre to each side.
First, we use the formula for the circumradius R given by:
[ R = \frac{abc}{4K} ]
where K is the area of the triangle. Using Heron's formula, we can find K:
[ s = \frac{a + b + c}{2} = \frac{8 + 6 + 7}{2} = 10.5 ]
[ K = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{10.5 \times 2.5 \times 4.5 \times 3.5} ]
[ K \approx 20.33 ]
Now, we can find R:
[ R = \frac{8 \times 6 \times 7}{4 \times 20.33} \approx 4.14 ]
Using the distances formulas, assuming we know the angles A, B, and C (which can be calculated using the Law of Cosines), we can find d_a, d_b, and d_c.
If A = 60°, B = 50°, and C = 70°, then:
[ d_a = R \cos A \approx 4.14 \cos 60° \approx 2.07 ] [ d_b = R \cos B \approx 4.14 \cos 50° \approx 2.67 ] [ d_c = R \cos C \approx 4.14 \cos 70° \approx 1.42 ]
These distances are the perpendicular distances from the circumcentre to the respective sides of the triangle.
Conclusion
The distance of the circumcentre from the sides of a triangle is a useful measure in various geometric problems and constructions. By understanding the properties of the circumcentre and using the appropriate formulas, one can calculate these distances efficiently. Remember that the circumcentre's position relative to the triangle (inside, on, or outside) depends on the type of triangle (acute, right, or obtuse), which affects the interpretation of these distances.