Problem 30 - Algebra 2

Question

GEOMETRY Circle \(Q\) has a diameter \(\overline{A B}\) . If \(A\) is at \((-3,-5)\) and the center of the circle is at \((2,3),\) find the coordinates of \(B\) .

Step-by-Step Solution

Answer

The coordinates of point B are (7, 11).

Step 1: Determine the Diameter Vector

The diameter of the circle is \(\overline{AB}\), which means that the center \((2,3)\) is the midpoint of the segment \(A (-3,-5)\) and \(B (x,y)\). We'll use the midpoint formula to find the coordinates of \(B\).

Step 2: Use the Midpoint Formula

The midpoint formula for a segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is given by:\( M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \). In this scenario, \(M = (2,3)\), \((x_1, y_1) = (-3, -5)\), and \((x_2, y_2) = (x, y)\).

Step 3: Set Up the Equations

Using the midpoint formula, we have two equations: * For the x-coordinates: \(\frac{-3 + x}{2} = 2\)* For the y-coordinates: \(\frac{-5 + y}{2} = 3\)

Step 4: Solve for x-Coordinate of B

Solve the first equation for \(x\):1. Multiply both sides by 2: \(-3 + x = 4\)2. Solve for \(x\): \(x = 7\) Thus, the x-coordinate of B is 7.

Step 5: Solve for y-Coordinate of B

Solve the second equation for \(y\):1. Multiply both sides by 2: \(-5 + y = 6\)2. Solve for \(y\): \(y = 11\) Thus, the y-coordinate of B is 11.

Step 6: Conclude the Coordinates of Point B

From the above calculations, the coordinates of point \(B\) are \((7, 11)\).

Key Concepts

Circle Properties

A circle is one of the most fundamental shapes in geometry, defined as the set of all points in a plane that are equidistant from a given point, known as the center. The fixed distance from the center to any point on the circle is called the radius. When we refer to the diameter, we speak of the straight line passing through the center, connecting two points on the circle's edge. The diameter is exactly twice the length of the radius.

Circles have interesting properties that make them unique:

• Every line through the center bisects the circle into two equal halves, known as semicircles.
• All diameters of the circle are of equal length, meaning they are congruent.
• The circle's circumference, which is the total distance around the circle, is calculated as \(2\pi r\), where \(r\) is the radius.

These basic properties are essential to solving geometry problems involving circles, such as locating specific points or understanding relationships within the circle.

Midpoint Formula

The midpoint formula is a handy tool in geometry for finding the exact middle point between two given endpoints of a line segment. Especially in coordinate geometry, this formula helps when dealing with figures like circles and triangles where symmetry often comes into play.

The formula is expressed as:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the endpoints. The values \( \frac{x_1 + x_2}{2} \) and \( \frac{y_1 + y_2}{2} \) represent the x and y coordinates of the midpoint respectively.

Using the midpoint formula, we can find critical information such as the location of the center of a circle if given two endpoints on its diameter. In our problem, since the center is known, it helps us find the unknown endpoint of the diameter.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves studying geometry using a coordinate system. This branch of mathematics connects algebra to geometry through graphs and equations.

By using coordinate geometry:

• You can describe geometric figures algebraically.
• The relationships between different geometric properties can be expressed with equations.

In our problem, coordinate geometry allows us to precisely define the position of points within a circle on a plane. It helps us use algebraic methods, such as solving equations, to find unknown values of coordinates. The beauty of coordinate geometry lies in its capability to provide a clear geometric interpretation of algebraic expressions, making complex geometric problems accessible.

Diameter

The diameter of a circle is a special line segment stretching from one point on the circle, through the center, to another point directly opposite it. As the longest possible chord in the circle, its significance is multifold.

Here are some key aspects of the diameter:

• The length of the diameter is twice that of the radius, mathematically represented as \(d = 2r\).
• It divides the circle into two equal parts or semicircles.
• Every point on the diameter is collinear with the center and that the midpoint of the diameter is actually the center of the circle.

In this exercise, knowing the diametric property helps utilize the midpoint formula to identify unknown points, enabling us to solve for the coordinates of missing points like B, given the center and another endpoint of the diameter.