Coordinate Geometry, Basic Introduction, Practice Problems

Name: Coordinate Geometry, Basic Introduction, Practice Problems
Uploaded: 2018-01-08T17:00:02.000Z
Duration: 1 h 6 min 2 s

Coordinate Geometry Problems

Finding Coordinates of Point B

The coordinates of point B can be determined by observing that it shares the same x-coordinate as point A due to a vertical line, which is 1. Additionally, it shares the same y-coordinate as point C because of a horizontal line, which is 2.

Calculating Area of a Right Triangle

To find the area of a right triangle, use the formula: Area = 1/2 * base * height. Here, the base (distance between points B and C) is calculated as 5 - 1 = 4 units, while the height (distance between points A and B) is found to be 5 - 2 = 3 units. Thus, Area = 1/2 * 4 * 3 = 6 square units.

Midpoint Formula for Circle Center

The center of a circle defined by endpoints A (1,2) and B (7,10) can be found using the midpoint formula: Midpoint M = ((x1 + x2)/2 , (y1 + y2)/2). This results in M being at (4,6).

Radius Calculation Using Distance Formula

To calculate the radius of the circle from its center to point C (7,10), apply the distance formula: r = √((x2 - x1)² + (y2 - y1)²). Substituting values gives r = √((7 - 4)² + (10 - 6)²) = √(9 +16) = √25 = 5 units.

Area and Circumference of Circle

The area of the circle is computed as Area = πr² → π(5²) → 25π square units. The circumference is given by Circumference = 2πr → 10π units. These formulas are essential for understanding circular geometry properties.

Standard Equation of a Circle

Deriving Standard Equation

The standard equation for a circle with center at (h,k) and radius r is expressed as: (x-h)² + (y-k)² = r². For this case with h=4 and k=6 and r=5:

(x - 4)² + (y - 6)² = 25 represents our circle's equation.

Tangent Line to Circle

Understanding Tangent Lines

A tangent line touches a circle at one point; here segment AB represents the radius meeting this tangent perpendicularly at point B on circle A's edge.

Slope Calculation Between Points A and B

To find the slope m between points A(3,2) and B(5,8), use m=(y₂-y₁)/(x₂-x₁): m=(8-2)/(5-3)=6/2=3.

Writing Equation for Line AB

Using point-slope form with known slope m and point A(3,2):

y - y₁=m(x-x₁), leads to:

y - 2 = 3(x -3). Distributing gives us y =3x -9 +2 → y =3x -7.

Finding Slope of Tangent Line

Since tangent lines are perpendicular to radii at their contact points, we need to determine their slopes based on negative reciprocals; thus if slope AB is m then slope tangent line l will be (-1/m).

This structured approach provides clarity on coordinate geometry concepts discussed in this video while allowing easy navigation through timestamps for further review or study.

Understanding Perpendicular and Tangent Lines

Finding the Slope of a Perpendicular Line

To find the slope of a perpendicular line, flip the fraction of the original slope and change its sign. For example, if the slope is 3/1 , it becomes -1/3 for the perpendicular line.

The slope of two parallel lines remains constant, while for perpendicular lines, it is defined as the negative reciprocal of the original slope.

Writing an Equation from a Point and Slope

When given a point not on a tangent line (point A), use another point on the tangent line (point B) to write an equation using either point-slope or slope-intercept formulas.

In this case, substituting values into the point-slope formula involves replacing y with 8 and m with -1/3 . Solve for b after eliminating fractions by multiplying through by 3.

Deriving Linear Equations

After calculations, derive that b = 29/3 . Thus, in slope-intercept form, the equation becomes:

y = -1/3x + 29/3 .

To convert to standard form ( Ax + By = C ), multiply through by 3 leading to:

x + 3y = 29 . This shows multiple ways to express linear equations.

Calculating Area Under Graph

Area Bounded by Axes and Line

The area bounded by axes and graph represented by x + y = 8 can be visualized as a triangle formed with intercept points calculated first.

Calculate intercepts:

Y-intercept occurs when x = 0; y = 8, giving coordinates (0,8).

X-intercept occurs when y = 0; x = 2, yielding coordinates (2,0). These points help in graphing.

Triangle Area Calculation

The area of this right triangle can be computed using:

Area formula:

Area = 1/2 * base * height.

Here base is 2 and height is 8 leading to an area of:

Area = 1/2 * 2 * 8 = 8 square units.

Plotting Points in Three Dimensions

Plotting Point Coordinates

To plot point (3,4,5):

Move three units along x-axis then four units along y-axis before moving up five units along z-axis.

Visualize these movements on respective axes for accurate plotting representation.

Calculating Distance from Origin

Use distance formula between two points in three dimensions:

Formula:

Distance = √((x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2).

For origin (0,0,0) to point P(3,4,5): calculate as follows:

Resulting calculation yields distance as approximately equal to five times square root two or about seven point zero seven units.

Distance Between Point and Line

Finding Perpendicular Distance from Point to Line

Given point (5,3) and line represented as equation format (ax+by+c=0), apply distance formula:

Distance formula:

d = |Ax_1 + By_1 + C| / √(A^2+B^2).

– Substitute values where A is coefficient of x from line equation; B is coefficient of y; C is constant term adjusted accordingly based on provided data points.

Example Calculation Steps

Area Calculation of Shaded Region

Understanding the Geometry

The distance from a point to a line is established as 4 units. The square defined by points A, B, C, and D has an area that needs to be calculated in relation to a circle with x-intercepts at (4, 0) and (-4, 0), and y-intercepts at (0, 4) and (0, -4).

Area of Square vs. Circle

To find the area of the shaded region, subtract the area of the circle from that of the square. The radius of the circle is determined to be 4 units; thus its diameter is 8 units. This also represents one side length of the square.

Calculating Areas

The area of the square is calculated as 8^2 = 64, while the area of the circle is given by pi r^2 = pi(4^2) = 16pi. Therefore, the exact area for the shaded region becomes 64 - 16pi.

Decimal Conversion

For practical purposes, converting 16pi into decimal form yields approximately 50.2655, leading to a final shaded area value around 13.7345 square units when expressed in decimal format.

Equilateral Triangle Area Calculation

Area Determination

An equilateral triangle ABC has point C at coordinates (a, 0). Given that AC measures eight units in length (from point A at (0,0)), we can calculate its area using fracsqrt34 s^2, where s equals side length. Thus for this triangle:

[

Area = fracsqrt34 times 8^2 = 16sqrt3

]

square units.

Finding Coordinates for Point B

Angle Analysis

In an equilateral triangle where all angles are congruent at 60^circ, splitting it creates two right triangles with angles measuring 90^circ -30^circ =60^circ. This leads us to analyze triangle ABD where AB serves as hypotenuse measuring eight units long.

Using Right Triangle Properties

Utilizing properties from a 30-60-90 triangle:

Side opposite 30^circ: half hypotenuse → Length: 4.

Side opposite 60^circ: equals half hypotenuse times sqrt3: Length: 4sqrt3.

Thus coordinates for point B are determined as (4, 4sqrt3).

Equation of Median in Triangle ABC

Median Definition

To write an equation for median BM extending from vertex B to midpoint M on segment AC requires finding M's coordinates first through midpoint formula:

[

M(x,y)= left( x_1+x_2/2, y_1+y_2/2 right).

]

With points A(1,2) and C(7,4), M calculates to (4,3).

Slope Calculation

Slope between points B(5,10) and M(4,3):

[

Slope(m)= Delta y/Delta x= m=10−3/5−4=7.

]

Using point-slope form gives us:

[

y−y_1=m(x−x_1).

]

Substituting values results in equation:

[

y = 7x −25.

]

This represents median BM's equation within triangle ABC.

Perpendicular Bisector Equation

Perpendicular Bisector Characteristics

The perpendicular bisector intersects segment AC at midpoint M(4,3), creating two equal segments while forming right angles with AC itself—this does not necessarily pass through vertex B unless specific conditions apply such as being part of an equilateral triangle setup.

What is the Slope of the Perpendicular Bisector?

Calculating the Slope of Line AC

The slope of line AC is calculated using points A (1, 2) and C (7, 4). The formula used is (y_2 - y_1) / (x_2 - x_1).

Substituting values gives a slope of (4 - 2) / (7 - 1) = 2 / 6, which simplifies to 1/3.

Finding the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector, denoted as line l, is the negative reciprocal of line AC's slope. Thus, it becomes -3/1.

Writing the Equation for Line L

Using point-slope form with point (4, 3) and slope -3, we calculate b:

y = mx + b

After calculations: b = 15.

Therefore, the equation for the perpendicular bisector is y = -3x + 15.

Writing the Equation for Altitude from Vertex B

The altitude from vertex B (5,10) to line AC also has a slope of -3. This altitude does not pass through midpoint but rather directly from B.

Finalizing Altitude Equation

Using point-slope form again:

With point B(5,10), we derive:

y = -3x + 25.