Questão de Geometria Analítica - equação da reta

Summary

In this tutorial, the speaker explains how to calculate the equation of a line based on the properties of perpendicular lines in a Cartesian coordinate system. Using the point of intersection of the diagonals of a rhombus, the speaker demonstrates how the slope of one diagonal's line is the negative reciprocal of the other. The point-slope formula is applied to find the equation of the second diagonal, ultimately revealing the correct equation as 2y + x + 1 = 0, corresponding to one of the given alternatives. The process highlights key concepts of geometry and algebra in a straightforward, step-by-step manner.

Takeaways

• 😀 The problem involves a Cartesian coordinate system and the intersection point P(-3, 1) of the diagonals of a rhombus.
• 😀 The diagonals of a rhombus are perpendicular, forming a 90-degree angle between them.
• 😀 If two lines are perpendicular, the slope (or coefficient angular) of one line is the opposite reciprocal of the other.
• 😀 The equation of one diagonal is given as Y = 2x - 2, where the slope is 2.
• 😀 The slope of the second diagonal must therefore be -1/2, as it is the opposite reciprocal of 2.
• 😀 The point P(-3, 1) is used to find the equation of the second diagonal.
• 😀 The formula to use is Y - y0 = m(X - x0), where (x0, y0) is the known point, and m is the slope of the line.
• 😀 Substituting the values from the given point (-3, 1) and the slope (-1/2) into the formula helps generate the equation of the second diagonal.
• 😀 The distributive property is used to simplify the equation, leading to an expression that needs to be arranged and equalized to zero.
• 😀 After simplifying and rearranging the terms, the final equation of the second diagonal becomes 2Y + X + 1 = 0.
• 😀 The correct answer corresponds to an alternative that matches this final equation, despite the order of terms being different.

Q & A

What is the main topic of the script?

The script discusses the analysis of a system of two perpendicular lines and the calculation of the equation of one of the diagonals of a rhombus in a Cartesian coordinate system.

What is meant by 'perpendicular lines' in the script?

Perpendicular lines are lines that intersect at an angle of 90 degrees, meaning their slopes are negative reciprocals of each other.

How do you calculate the slope of a perpendicular line based on the given information?

The slope of the second perpendicular line is the negative reciprocal of the slope of the first line. If the first slope is m1, the second slope m2 would be -1/m1.

What is the equation of the first diagonal line provided in the script?

The equation of the first diagonal is given as y = (1/2)x - 2, where the coefficient of x (1/2) represents the slope of the line.

How do you find the equation of the second diagonal line?

Using the point (-3, 1) and the negative reciprocal slope of -1/2, you substitute these values into the point-slope form to find the equation of the second diagonal line.

What is the point-slope form equation used in this problem?

The point-slope form equation used is y - y0 = m(x - x0), where m is the slope and (x0, y0) is the given point.

What values are substituted into the point-slope form to solve the equation?

The point (-3, 1) is used, where x0 = -3 and y0 = 1, and the slope m = -1/2 is substituted into the equation.

How is the equation simplified after substitution?

After substitution, the equation y - 1 = -1/2(x + 3) is expanded and simplified to y = -1/2x + 1/2 + 3, resulting in the equation 2y + x - 2 = 0.

What does the final simplified equation represent?

The final equation, 2y + x - 2 = 0, represents the equation of the second diagonal line, derived from the point (-3, 1) and the slope -1/2.

How are the alternatives presented in the problem relevant?

The alternatives present variations of the equation, and despite differences in order, the correct answer remains the same. The equation 2y + x - 2 = 0 is equivalent to the alternative provided as letter C.