Problem 70 - Calculus Early Transcendentals: Pearson New International Edition

Question

The center of the circumscribed circle of a triangle lies on the perpendicular bisectors of the sides. Use this fact to find the center of the circle that circumscribes the triangle with vertices \((0,4),(2,0)\), and \((4,6)\)

Step-by-Step Solution

Answer

The center of the circle is approximately \((2, 3.33)\).

Step 1: Find the Midpoint of Each Side

Calculate the midpoints of the segments connecting the given vertices of the triangle. The first side is between points \((0,4)\) and \((2,0)\). The midpoint is: \(\left( \frac{0+2}{2}, \frac{4+0}{2} \right) = (1, 2)\). For the second side between points \((2,0)\) and \((4,6)\), the midpoint is \(\left( \frac{2+4}{2}, \frac{0+6}{2} \right) = (3, 3)\). For the third side between points \((4,6)\) and \((0,4)\), the midpoint is \(\left( \frac{4+0}{2}, \frac{6+4}{2} \right) = (2, 5)\).

Step 2: Determine the Slope of Each Side

Calculate the slopes of the lines between each pair of vertices. Between points \((0,4)\) and \((2,0)\), the slope \(m_1 = \frac{0-4}{2-0} = -2\). Between points \((2,0)\) and \((4,6)\), the slope \(m_2 = \frac{6-0}{4-2} = 3\). Between points \((4,6)\) and \((0,4)\), the slope \(m_3 = \frac{4-6}{0-4} = \frac{1}{2}\).

Step 3: Find the Equations of the Perpendicular Bisectors

The perpendicular bisector of a line segment has a slope that is the negative reciprocal of the slope of the segment. For the first side, the perpendicular bisector slope is \(\frac{1}{2}\) and the equation using point \((1,2)\) is \(y - 2 = \frac{1}{2}(x - 1)\). For the second side, the perpendicular slope is \(-\frac{1}{3}\) and the equation using point \((3,3)\) is \(y - 3 = -\frac{1}{3}(x - 3)\). For the third side, the perpendicular slope is \(-2\) and the equation using point \((2,5)\) is \(y - 5 = -2(x - 2)\).

Step 4: Solve the Perpendicular Equations to Find the Center

Solve the equations of the perpendicular bisectors to find their intersection point. First solve \(y - 2 = \frac{1}{2}(x - 1)\) and \(y - 3 = -\frac{1}{3}(x - 3)\). Set them equal: \(2 + \frac{1}{2}(x - 1) = 3 - \frac{1}{3}(x - 3)\). Solve for \(x = 2\). Substitute \(x = 2\) into one equation, \(y - 3 = -\frac{1}{3}(2 - 3)\), so \(y = \frac{1}{3} + 3 = 3.33\). Thus the center is approximately \((2, 3.33)\).

Key Concepts

Perpendicular Bisectors

A perpendicular bisector is a line that divides a line segment into two equal parts at a 90-degree angle. In the geometry of triangles, the perpendicular bisectors of a triangle's sides intersect at a single point, known as the circumcenter, which is the center of the circumscribed circle.
This circle passes through all three vertices of the triangle.

• The slope of the perpendicular bisector is the negative reciprocal of the slope of the original segment.
• For example, if a segment has a slope of 2, the perpendicular bisector will have a slope of \(-1/2\).

The equations of the perpendicular bisectors can be used to find their intersection point through solving simultaneous equations, pinpointing the circumcenter of the triangle.

Midpoint of a Line Segment

The midpoint of a line segment is a point that divides the segment into two equal lengths. To find this midpoint, take the average of the x-coordinates and the y-coordinates of the endpoints. This can be expressed as:
\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]

• It's important for constructing perpendicular bisectors because the bisector passes through this midpoint.
• For example, the segment between points \(0,4\) and \(2,0\) has a midpoint at \( (1, 2) \).

The midpoint aids in determining crucial intersections needed for geometrical constructions within a triangle.

Slope of a Line

The slope of a line determines its steepness and direction. It is calculated by the rise over run, or the change in y divided by the change in x:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• A positive slope means the line rises as it moves right, while a negative slope means it falls.
• A zero slope indicates a horizontal line, and an undefined slope indicates a vertical line.

The concept of slope is crucial in finding perpendicular bisectors since the negative reciprocal of a line's slope gives the slope of its perpendicular bisector. This facilitates the process of determining the equations necessary for finding intersecting points like the center of a circumscribed circle.

Geometry of Triangles

Triangles are fundamental shapes in geometry with diverse properties and types, including equilateral, isosceles, and scalene. A triangle's geometry involves important aspects like its angles, sides, and intersection points.

• The intersection of the perpendicular bisectors is called the circumcenter; it's equidistant from all the vertices of the triangle.
• This knowledge is vital when drawing or calculating aspects of triangles and circles in geometrical problems.

Understanding the properties of triangles allows us to apply facts such as that the circumcenter is the same distance from each vertex, which helps solve various complex problems, involving shapes like the circumscribed circle.