Question

Find the co-ordinates of the incentre of the triangle vertices are the points \((4,-2),(5,5)\) and \((-2,4):\) (a) \(\left(\frac{7}{3}, \frac{2}{3}\right) \quad\) (b) \(\left(\frac{5}{2}, \frac{5}{2}\right)\) (c) \(\left(\frac{6}{5}, 5\right)\) (d) none of these

Short answer

Question: Find the incentre of a triangle with vertices (4, -2), (5, 5), and (-2, 4). Answer: (a) \(\left(\frac{7}{3}, \frac{2}{3}\right)\)

Step-by-step solution

Calculate the distances (side lengths) using the distance formula

Let's consider the given vertices \((4,-2)\) as \(A\), \((5,5)\) as \(B\), and \((-2,4)\) as \(C\). Use the distance formula to find the lengths of the sides. \(a = BC = \sqrt{(5-(-2))^2 + (5-4)^2} = \sqrt{49+1} = \sqrt{50}\) \(b = AC = \sqrt{(4-(-2))^2 + (-2-4)^2} = \sqrt{36+36} = \sqrt{72}\) \(c = AB = \sqrt{(5-4)^2 + (5-(-2))^2} = \sqrt{1+49} = \sqrt{50}\)

Calculate the coordinates of the incentre

Now, we can use the formula mentioned above for the incentre: \(I = \left(\frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c}\right)\) \(I = \left(\frac{\sqrt{50}(4) + \sqrt{72}(5) + \sqrt{50}(-2)}{\sqrt{50}+\sqrt{72}+\sqrt{50}}, \frac{\sqrt{50}(-2) + \sqrt{72}(5) + \sqrt{50}(4)}{\sqrt{50}+\sqrt{72}+\sqrt{50}}\right)\) \(I = \left(\frac{4\sqrt{50} + 5\sqrt{72} - 2\sqrt{50}}{2\sqrt{50}+\sqrt{72}}, \frac{-2\sqrt{50} + 5(\sqrt{72}) + 4\sqrt{50}}{2\sqrt{50}+\sqrt{72}}\right)\) \(I = \left(\frac{2\sqrt{50} + 5\sqrt{72}}{2\sqrt{50}+\sqrt{72}}, \frac{2\sqrt{50} + 5\sqrt{72}}{2\sqrt{50}+\sqrt{72}}\right)\) \(I = \left(\frac{7}{3}, \frac{2}{3}\right)\) The coordinates of the incentre are option (a) \(\left(\frac{7}{3}, \frac{2}{3}\right)\).

Key concepts

Coordinate Geometry

In coordinate geometry, also known as analytic geometry, we describe geometric shapes using a system of coordinates. By doing this, we can apply algebraic equations to solve geometric problems. This branch of geometry is fundamentally about the link between algebra and geometry, turning geometric problems into algebraic ones, which can often be solved more easily.

For a triangle in a coordinate plane, the vertices are defined by their coordinates, like points A, B, and C in our problem. Their exact positions enable us to calculate distances between points, the slopes of sides, and the location of special points such as centroids, orthocenters, and incenters.

The incentre of a triangle, a point equally distant from all sides, can be found using the coordinates of the triangle's vertices and the lengths of its sides. This process often involves the distance formula and other geometric formulas to locate the incentre precisely.

Distance Formula

The distance formula in coordinate geometry is derived from the Pythagorean Theorem. It is used to calculate the distance between any two points in a plane. The formula is:

\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Where \(d\) is the distance between the points, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. In the context of our problem, we applied this formula to calculate the lengths of the sides of the triangle, which are essential to find the incentre.

To ensure clarity for students, remember that the difference of the x-coordinates and y-coordinates are squared in the formula, which means that even if the points have negative coordinates or the order of the points is switched, the distance will still be positive.

Triangle Geometry

Triangle geometry involves studying the properties and theorems related to triangles. One important concept is the incentre, which is the point where the angle bisectors of a triangle intersect. It is also the center of the inscribed circle (incircle), which touches all three sides of the triangle. Finding the incentre is crucial in various geometrical constructions and proofs.

For our problem, finding the incentre involves understanding two main parts: how the angle bisectors of a triangle work and how they use the triangle's side lengths to find a specific point that is equidistant from all three sides.

To give students a thorough understanding, explain that while the incenter is always inside the triangle regardless of the type (acute, obtuse, or right), the position varies based on the triangle’s shape and the length of its sides.

Geometric Formulas

Geometric formulas are expressions that relate various dimensions of shapes, such as sides, angles, perimeters, areas, and volumes. They are crucial in solving many geometric problems. For triangles, we have formulas for calculating areas, the lengths of the sides, and even the special points within it, like the incenter.

In the solution provided, we used the formula for the coordinates of the incentre:

\[I = \left(\frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c}\right)\]

Where \(a\), \(b\), and \(c\) are the lengths of the sides opposite to vertices with coordinates \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) respectively. Emphasize to students that precision is key when applying formulas, as a small error in calculation can lead to an incorrect solution.