Perpendicular Lines and Equations of Straight Lines

This page delves into the properties of perpendicular lines and the methods for finding equations of straight lines.

Key concepts covered include:

• Gradients of perpendicular lines
• General equation of a straight line
• Point-slope form of a line equation

Vocabulary: Perpendicular lines are lines that intersect at right angles (90 degrees).

Formula: The equation of a straight line formula is y = mx + c, where m is the gradient and c is the y-intercept.

Highlight: For perpendicular lines, the product of their gradients is always -1.

Example: Given points A(4,7) and B$3,-10$, the gradient of line AB is calculated as m = $7-(-10)$/$4-3$ = 17.

The page also demonstrates how to find the equation of a line using the point-slope form: y - y₁ = m$x - x₁$, where (x₁, y₁) is a point on the line and m is the gradient.

Example: For line CD with gradient -1/3 passing through point D(1,5), the equation is derived as: y - 5 = -1/3$x - 1$, which simplifies to x + 3y - 16 = 0.

Perpendicular Bisectors

This page focuses on perpendicular bisectors and their applications in triangle geometry.

The main topics covered are:

• Finding the equation of a perpendicular bisector
• Types of lines in triangles (altitudes, medians)
• Handling vertical and horizontal lines

Definition: A perpendicular bisector is a line that passes through the midpoint of a line segment at a right angle.

Example: To find the equation of perpendicular bisector of line AB where A$5,-1$ and B(13,23), first calculate the midpoint (9,11), then use the perpendicular gradient to form the equation: y - 11 = -3/8$x - 9$.

Highlight: When dealing with vertical lines, the gradient is undefined, and the equation takes the form x = constant.

The page also introduces the concept of altitudes in triangles, which are perpendicular lines from a vertex to the opposite side.

Vocabulary: An altitude of a triangle is a line segment from a vertex perpendicular to the line containing the opposite side (or its extension).

Medians and Triangle Centers

This page explores medians in triangles and various triangle centers.

Key concepts include:

• Finding the equation of a median
• Locating the centroid of a triangle
• Relationships between different triangle centers

Definition: A median is a line segment that connects a vertex to the midpoint of the opposite side in a triangle.

Example: In triangle ABC with A$-5,6$, B$-4,-10$, and C(3,12), to find the equation of median BD, first calculate the midpoint of AC $-1,9$, then use the point-slope form with B$-4,-10$ to get the equation: 3y = 19x + 46.

Highlight: The centroid of a triangle is the point where all three medians intersect.

The page also mentions other important triangle centers:

• Orthocenter: The point where all three altitudes intersect
• Circumcenter: The point where all three perpendicular bisectors of the sides intersect

Advanced Line Problems and Triangle Centers

This final page covers more complex problems involving lines and triangle centers.

Topics addressed include:

• Finding equations of lines making specific angles with the x-axis
• Solving for intersection points of perpendicular bisectors
• Locating and understanding the significance of triangle centers

Example: To find the equation of a line AB making a 45° angle with the positive x-axis direction, use m = tan 45° = 1, resulting in the equation y = x - 3.

Highlight: The intersection point of perpendicular bisectors in a triangle is significant as it represents the circumcenter, which is equidistant from all three vertices.

The page concludes with a problem involving finding the intersection point of two lines, demonstrating the use of simultaneous equations to solve for the coordinates (8,5).

Vocabulary: The centroid of a triangle divides each median in the ratio 2:1, with the centroid being closer to the midpoint of the side.

This comprehensive guide provides a thorough understanding of how to find the equation of a straight line with one point, how to find midpoint coordinates on a graph, and various aspects of perpendicular bisector equations and calculations.

Collinearity and Distance

This page introduces fundamental concepts in coordinate geometry, including collinearity and distance between points.

The page covers:

• Determining if points are collinear
• Calculating the distance between two points using the distance formula
• Finding the midpoint of a line segment

Definition: Collinearity refers to points that lie on the same straight line.

Formula: The distance formula is used to calculate the length of a line segment: d = √$(x₂-x₁)² + (y₂-y₁)²$

Example: For points A(1,7) and B$5,-3$, the distance AB is calculated as √$(5-1)² + (-3-7)²$ = √$16 + 100$ = √116 ≈ 10.77 units.

Highlight: The midpoint formula is crucial for finding the coordinates of the point exactly in the middle of a line segment: $x₁+x₂$/2, $y₁+y₂$/2

The page also touches on the concept of gradient and its relationship to the angle a line makes with the x-axis, introducing the formula m = tan θ.