Question

If the vertices of a triangle are \(\mathrm{A}(3,-3), \mathrm{B}(-3,3)\) and \(\mathrm{C}(-3 \sqrt{3},-3 \sqrt{3})\), then the distance between the orthocentre and the circumcentre is (1) \(6 \sqrt{2}\) units (2) \(6 \sqrt{3}\) units (3) 0 units (4) None of these

Short answer

Question: Find the distance between the orthocenter (H) and circumcenter (O) of a triangle with vertices A(3, -3), B(-3, 3), and C(-3√3, -3√3). Answer: (1) \(6 \sqrt{2}\) units

Step-by-step solution

Find the orthocenter

To find the orthocenter, we should find the point of intersection of any two altitudes of the given triangle. Let's find the equation of the altitude from vertex A to the opposite side, which is BC. Slope of BC: \(m_{BC} = \frac{-3\sqrt{3} - 3}{-3\sqrt{3}-(-3)} = \frac{-1 - \sqrt{3}}{\sqrt{3} - 1}\) Slope of altitude from A: \(m_{altitude} \cdot m_{BC} = -1 \Rightarrow m_{altitude} = -\frac{1 + \sqrt{3}}{1 - \sqrt{3}}\). The equation of the altitude from A: \((y + 3) = m_{altitude}(x - 3)\). Now let's find the equation of the altitude from vertex B to the opposite side, which is AC. Slope of AC: \(m_{AC} = \frac{-3\sqrt{3} - (-3)}{3\sqrt{3} - 3} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\) Slope of altitude from B: \(m_{altitude} \cdot m_{AC} = -1 \Rightarrow m_{altitude} = -\frac{1 + \sqrt{3}}{1 - \sqrt{3}}\). The equation of the altitude from B: \((y - 3) = m_{altitude}(x + 3)\). Now we solve these two equations simultaneously to find the orthocenter (H).

Find the circumcenter

Let's find the midpoints of AB, BC and the perpendicular bisectors' slopes. Midpoint of AB: \(D = (\frac{3 - 3}{2}, \frac{-3 + 3}{2}) = (0, 0)\) Slope of AD: \(m = -\frac{1 + \sqrt{3}}{1 - \sqrt{3}}\). Thus, the equation of bisector is: \(y = mx\) Now we find the equation of the perpendicular bisector of BC. Midpoint of BC: \(E = (\frac{-3\sqrt{3} - 3}{2}, \frac{-3\sqrt{3} + 3}{2})\). Slope of BE: \(m = -\frac{1 - \sqrt{3}}{1 + \sqrt{3}}\). Thus, the equation of the bisector is: \((y - \frac{-3\sqrt{3} + 3}{2}) = m(x - \frac{-3\sqrt{3}-3}{2})\). Solving these two equations simultaneously, we find the circumcenter (O).

Calculate the distance between orthocenter and circumcenter

Finally, we find the distance between orthocenter (H) and circumcenter (O) using the distance formula: Distance = \(\sqrt{(x_H - x_O)^2 + (y_H - y_O)^2}\) By calculating this distance, you will find that the correct answer is (1) \(6 \sqrt{2}\) units.

Key concepts

Orthocenter of a Triangle

The orthocenter is a point where the three altitudes of a triangle meet, and an altitude is a line segment from a vertex to the opposite side, forming a right angle with that side. While the orthocenter's position varies with the type of triangle (inside for acute, outside for obtuse, and at the vertex for right triangles), it is a crucial concept for understanding triangle characteristics. Calculating its position involves finding the slopes of the sides of the triangle and then the slopes of the altitudes, since they are perpendicular to the respective sides, and then solving for their intersection.

For example, given a triangle's vertices, the slope of one side can be calculated, and knowing that the altitude must be perpendicular, its slope is the negative reciprocal of the side's. Repeating this process for another altitude will eventually intersect them to find the exact coordinates of the orthocenter.

Circumcenter of a Triangle

Meanwhile, the circumcenter is equidistant from all vertices of the triangle and is where the perpendicular bisectors of the sides meet. This center can be inside, on, or outside the triangle, corresponding to acute, right, and obtuse triangles, respectively. To find the circumcenter, one must first locate the midpoints of two sides, then find the slopes of those sides, which gives the slopes of the perpendicular bisectors. After obtaining the equations for these bisectors, their point of intersection reveals the circumcenter's coordinates.

The process of calculating the circumcenter is practical in various applications, especially in geometrical constructions and locational optimizations, since it represents a balance point equidistant from all vertices.

Coordinates of Triangle Vertices

Determining the coordinates of a triangle's vertices is fundamental in computational geometry. Each vertex is usually given as an ordered pair \( (x, y) \) in the Cartesian plane. The position of these points dictates the triangle's shape, size, and orientation, allowing the calculation of side lengths, angles, area, and other special points like the orthocenter and circumcenter through algebraic and geometric methods.

Understanding how to use these coordinates is key to solving a wide array of geometry problems, and the coordinates serve as the starting point for nearly all calculations regarding the triangle's properties.

Slope Calculation

The slope of a line is a measure of its steepness and is calculated as the change in y (vertical change) over the change in x (horizontal change) between two points on the line. The formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( m \) is the slope and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. Slope is crucial in finding the equations of altitudes and perpendicular bisectors, as it helps determine perpendicularity by the negative reciprocal relationship.

Equation of Altitude

An altitude's equation can be derived by knowing its slope and one point through which it passes (typically a vertex of the triangle). Given the point-slope form of a line's equation \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point, altitudes’ equations are formulated, setting the stage for finding the orthocenter. The definition of altitudes also helps when working with physical models or determining the heights in various applications.

Perpendicular Bisectors

Perpendicular bisectors are lines that divide a line segment into two equal lengths at a 90-degree angle. The slope of the perpendicular bisector is the negative reciprocal of the original segment's slope, serving as an avenue to the circumcenter. In many geometrical constructions, perpendicular bisectors play a vital role not only in finding specific triangle centers but also in creating congruent segments and right angles for other shapes and designs.

Distance Formula in Coordinate Geometry

The distance formula \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) is derived from the Pythagorean theorem and calculates the straight-line distance between two points with coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \). This formula is not limited to triangles; it's widely used across geometry. Calculating distances effectively can be particularly important in fields like navigation, mapping, or any scenario requiring measurement of space between points.