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(Geometry: intersecting point) Two points on line 1 are given as $(x 1, y 1)$ and $(x 2,$ $y 2)$ and on line 2 as $(x 3, y 3)$ and $(x 4, y 4),$ as shown in Figure $3.10 a-b$ The intersecting point of the two lines can be found by solving the following linear equation: \[\begin{array}{l}\left(y_{1}-y_{2}\right) x-\left(x_{1}-x_{2}\right) y=\left(y_{1}-y_{2}\right) x_{1}-\left(x_{1}-x_{2}\right) y_{1} \\\left(y_{3}-y_{4}\right) x-\left(x_{3}-x_{4}\right) y=\left(y_{3}-y_{4}\right) x_{3}-\left(x_{3}-x_{4}\right) y_{3}\end{array}\] This linear equation can be solved using Cramer's rule (see Exercise 3.3 ). If the equation has no solutions, the two lines are parallel (Figure $3.10 \mathrm{c}$ ). Write a program that prompts the user to enter four points and displays the intersecting point. Here are sample runs: Enter $x 1, \quad y 1, \quad x 2, y 2, x 3, y 3, x 4, y 4: 225-1.04 .0 \quad 2.0-1.0-2.0$ The intersecting point is at (2.88889,1.1111) Enter $x 1, y 1, x 2, y 2, x 3, y 3, x 4, y 4: 2276.04 .02 .0-1.0-2.0$ The two 1 ines are paralle?

   (Geometry: intersecting point) Two points on line 1 are given as $(x 1, y 1)$ and $(x 2,$
$y 2)$ and on line 2 as $(x 3, y 3)$ and $(x 4, y 4),$ as shown in Figure $3.10 a-b$
The intersecting point of the two lines can be found by solving the following linear equation:
\[\begin{array}{l}\left(y_{1}-y_{2}\right) x-\left(x_{1}-x_{2}\right) y=\left(y_{1}-y_{2}\right) x_{1}-\left(x_{1}-x_{2}\right) y_{1} \\\left(y_{3}-y_{4}\right) x-\left(x_{3}-x_{4}\right) y=\left(y_{3}-y_{4}\right) x_{3}-\left(x_{3}-x_{4}\right) y_{3}\end{array}\]
This linear equation can be solved using Cramer's rule (see Exercise 3.3 ). If the equation has no solutions, the two lines are parallel (Figure $3.10 \mathrm{c}$ ). Write a program that prompts the user to enter four points and displays the intersecting point. Here are sample runs:
Enter $x 1, \quad y 1, \quad x 2, y 2, x 3, y 3, x 4, y 4: 225-1.04 .0 \quad 2.0-1.0-2.0$
The intersecting point is at (2.88889,1.1111)
Enter $x 1, y 1, x 2, y 2, x 3, y 3, x 4, y 4: 2276.04 .02 .0-1.0-2.0$
The two 1 ines are paralle?

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Introduction to Java Programming

Introduction to Java Programming

Y. Daniel Liang 9th Edition

Chapter 3, Problem 25

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Solved by Expert Morgan Cheatham

09/21/2022

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First, we need to take the input from the user for the four points (x1, y1, x2, y2, x3, y3, x4, y4). Show more…

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(Geometry: intersecting point) Two points on line 1 are given as $(x 1, y 1)$ and $(x 2,$ $y 2)$ and on line 2 as $(x 3, y 3)$ and $(x 4, y 4),$ as shown in Figure $3.10 a-b$ The intersecting point of the two lines can be found by solving the following linear equation: \[\begin{array}{l}\left(y_{1}-y_{2}\right) x-\left(x_{1}-x_{2}\right) y=\left(y_{1}-y_{2}\right) x_{1}-\left(x_{1}-x_{2}\right) y_{1} \\\left(y_{3}-y_{4}\right) x-\left(x_{3}-x_{4}\right) y=\left(y_{3}-y_{4}\right) x_{3}-\left(x_{3}-x_{4}\right) y_{3}\end{array}\] This linear equation can be solved using Cramer's rule (see Exercise 3.3 ). If the equation has no solutions, the two lines are parallel (Figure $3.10 \mathrm{c}$ ). Write a program that prompts the user to enter four points and displays the intersecting point. Here are sample runs: Enter $x 1, \quad y 1, \quad x 2, y 2, x 3, y 3, x 4, y 4: 225-1.04 .0 \quad 2.0-1.0-2.0$ The intersecting point is at (2.88889,1.1111) Enter $x 1, y 1, x 2, y 2, x 3, y 3, x 4, y 4: 2276.04 .02 .0-1.0-2.0$ The two 1 ines are paralle?

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Key Concepts

Line Representation in Coordinate Geometry

A line in a plane can be represented by an equation derived from two distinct points. By determining the line's slope and using point-slope or standard form, one can transform the geometric description of a line into an algebraic equation.

Parallel Lines

Parallel lines in the coordinate plane have identical slopes, which leads to a situation where the system of equations has no unique solution because the determinant used in Cramer's Rule becomes zero. This condition signifies that the lines do not intersect at a single point.

System of Linear Equations

In coordinate geometry, the intersection between two lines is found by solving a system of linear equations. Each linear equation represents a line in the plane, and their solution gives the common point where both lines meet, if it exists.

Cramer's Rule

Cramer's Rule is a method that uses determinants to solve a set of linear equations. It provides a systematic way to find the values of the variables by dividing specific determinants, ensuring that a unique solution is obtained when the system's determinant is nonzero.

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