Problem 100 - A Graphical Approach to Precalculus with Limits

Question

Suppose that \(P\) is an endpoint of a segment \(P Q\) and \(M\) is the midpoint of \(P Q .\) Find the coordinates of endpoint \(Q\). $$P(-10.32,8.55), M(1.55,-2.75)$$

Step-by-Step Solution

Answer

The coordinates of endpoint \(Q\) are \((13.42, -14.05)\).

Step 1: Understanding the midpoint formula

The midpoint_between two points \(x_1, y_1\) and \(x_2, y_2\)\ uses the formula: \(M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\). In the given problem, we know the coordinates for \P\ and \M\ and need to find the_coordinates_for \Q\.\ Let's_use_the_midpoint_formula to set up_equations_for_the_unknown_coordinates_of_\Q\.

Step 2: Set up equations for x-coordinate

Given \(P_x = -10.32\) and \(M_x = 1.55\), the midpoint formula gives: (actually using \(M_x = \frac{P_x + Q_x}{2} \)): \[ 1.55 = \frac{-10.32 + Q_x}{2} \]. Solve this equation for \(Q_x\).

Step 3: Solve for x-coordinate of Q

Multiply both sides by 2 to eliminate the fraction: \[2 \times 1.55 = -10.32 + Q_x \]. Simplify and solve for \(Q_x\): \[3.10 = -10.32 + Q_x \Rightarrow Q_x = 3.10 + 10.32 \Rightarrow Q_x = 13.42\].

Step 4: Set up equations for y-coordinate

Use the given \(P_y = 8.55\), \(M_y = -2.75\), and the midpoint formula: (again, it's using \(M_y = \frac{P_y + Q_y}{2}\)) \[-2.75 = \frac{8.55 + Q_y}{2} \]. Solve this equation for \(Q_y\).

Step 5: Solve for y-coordinate of Q

Multiply both sides by 2 to clear the fraction: \[2 \times -2.75 = 8.55 + Q_y \]. Simplify and solve for \(Q_y\): \[-5.50 = 8.55 + Q_y \Rightarrow Q_y = -5.50 - 8.55 \Rightarrow Q_y = -14.05\].

Step 6: Write the coordinates of Q

With \(Q_x\) and \(Q_y\) determined, the endpoint \(Q\) is: \(Q(13.42, -14.05)\). This represents the coordinates of endpoint \(Q\) such that \M\ is the midpoint of \PQ\.

Key Concepts

Midpoint Formula

The midpoint formula is a useful tool in coordinate geometry, allowing us to find the exact middle point between two coordinates. This formula applies not only in classroom exercises but also in real-life situations. When you need to know the center point between two places, you can use this formula. For any segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \(M\) can be found using:

• \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)

The operation is simple: you calculate the average of the x-coordinates and y-coordinates separately. This gives you the midpoint's x and y values.
For instance, in the exercise, we're given point \(P(-10.32, 8.55)\) and midpoint \(M(1.55, -2.75)\). We can set up and solve equations to find the unknown endpoint \(Q\). Therefore, understanding and using the midpoint formula correctly can immensely simplify the process of determining point positions on a coordinate plane.

Endpoints

Endpoints are significant in determining segments in coordinate geometry. If you think of a line segment, it's confined from one endpoint to another. In the exercise, one endpoint, \(P\), and the midpoint, \(M\), are known. We need to find the missing endpoint, \(Q\).
Every midpoint divides a segment into two equal parts. This means both sub-segments must have the same length. Thus, possessing at least one endpoint and the midpoint ensures you can always find the other endpoint through the midpoint formula by rearranging the coordinates logically and solving equations.

• To solve for \(Q_x\), consider: \(M_x = \frac{P_x + Q_x}{2}\).
• To solve for \(Q_y\), consider: \(M_y = \frac{P_y + Q_y}{2}\).

Knowing how to utilize endpoints effectively can enable you to solve various geometry challenges accurately.

Solving Equations

Solving equations in coordinate geometry is often straightforward once the system of equations is set up. In our exercise, we used the known values from the midpoint and one endpoint to find the coordinates of the unknown endpoint. The process of solving these step-by-step ensures logical and error-free results.
For finding \(Q_x\), the equation was:

• \[1.55 = \frac{-10.32 + Q_x}{2}\]

Multiply by 2 to clear the fraction, then solve for \(Q_x\):

• \[3.10 = -10.32 + Q_x\]
• \[Q_x = 3.10 + 10.32 = 13.42\]

Similarly, for \(Q_y\):

• \[-2.75 = \frac{8.55 + Q_y}{2}\]

Multiply by 2, then solve:

• \[-5.50 = 8.55 + Q_y\]
• \[Q_y = -5.50 - 8.55 = -14.05\]

By breaking these equations down into simpler operations, you can confidently work through problems to find solutions. Remember, solving equations accurately is a fundamental skill in coordinate geometry and crucial for finding precise positional data.