# data[].attributes.body
Parallel lines have the SAME slope.

# data[].attributes.body
How do we find the perimeter and area of basic figures and irregular figures on the coordinate plane?

How do I use coordinate geometry to prove
geometric theorems?

How do I prove that four coordinate points form
a rectangle?

How do I use the coordinate plane to justify geometric relationships?

# data[].attributes.body
Students will compute the perimeter of regular and irregular figures in the coordinate plane.

Students will compute the area of triangles, rectangles, and irregular figures in the coordinate plane.

Students will write equations of parallel and perpendicular lines.

Student prove or disprove a claim about a figure.

# data[].attributes.body
PRIORITY:
G.GPE.7​ Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

SUPPORTING:
G.GPE.4​ Use coordinates to prove simple geometric theorems algebraically. For example, prove or
disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or
disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0,
2).

G.GPE.5​ Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes
through a given point).

# data[].attributes.attachment_link.title
Area and Perimeter of Irregular Shapes by Shmoop

# data[].attributes.attachment_link.description
The way to figure out the area and perimeter of irregular shapes is to break it down into regular shapes. A flower can be broken down into semi-circles and a square. Don't try to do it the long way, the flower may wilt in that time.

# data[].attributes.attachment_link.description
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

# data[].attributes.body
Area & perimeter on the coordinate plane

# data[].attributes.attachment_link.title
Area & perimeter on the coordinate plane (practice) | Khan Academy

# data[].attributes.attachment_link.description
Find the area or perimeter of shapes like triangles, rectangles, parallelograms, and hexagons on the coordinate plane.

# data[].attributes.body
Area of trapezoid on the coordinate plane

# data[].attributes.attachment_link.title
Area of trapezoid on the coordinate plane

# data[].attributes.attachment_link.description
Sal finds the area of a trapezoid using the distance formula and the trapezoid area formula.

# data[].attributes.body
Basic
Triangle
Rectangle
Square
Base
Formula
Quadrilaterals
Square units
Basic
Triangle
Rectangle
Square
Base
Formula
Height
Length
Width

# data[].attributes.body
Perimeter
Area
Irregular figure
Polygons
Height
Length
Width
Irregular figure
Polygons
Quadrilaterals
Coordinate plane
Ordered pair
Distance formula
Pythagorean theorem
Square root
Parallel
Perpendicular
Point-slope
Slope

# data[].attributes.attachment_link.title
Equal Area Triangles on the Same Base I-student

# data[].attributes.attachment_link.title
Equal Area Triangles on the Same Base II-student

# data[].attributes.body
The Slopes of Two Perpendicular Lines Are Opposite Reciprocals
of One Another

# data[].attributes.attachment_link.readable_text
Essential Question: How are the slope and the distance formula used to determine properties of polygons and circles?

Lesson

A figure in a coordinate plane may appear to be one of the familiar quadrilaterals, but proof is needed to show that the figure meets the definition of the particular shape. The slope formula and distance formula can be used to prove properties of quadrilaterals and to classify them.
Slope of a Line
The slope of a non-vertical line can be found using the formula below. The slope of a vertical line is undefined.

m(AB) is used to denote the slope of line segment AB.

For instance, find the slope of the line between (-3, 5) and (2, 1).
Distance Between Two Points
The distance between two points can be found using the formula below.

For instance, find the distance between (-2, 5) and (3, -1).
Perpendicular and Parallel Lines
To show that two lines are parallel, show that their slopes are equal.

To show that two lines are perpendicular (form a right angle), show that the product of their slopes is -1 (the slopes are opposite reciprocals). The symbol for perpendicular is ┴.

To show that two lines are congruent, show that they have equal measure (equal length) by using the distance formula.

Proving Quadrilaterals

To prove a quadrilateral is a parallelogram:
Show that both pairs of opposite sides are parallel.
Show that both pairs of opposite sides are congruent.

To prove a quadrilateral is a rhombus:
Show that all four sides are congruent.

To prove a quadrilateral is a rectangle:
First show it is a parallelogram by showing that both pairs of opposite sides are parallel, and then show that one angle is a right angle (the other 3 angles will be right angles due to other properties of parallelograms).
Show that all four angles are right angles.

To prove a quadrilateral is a square:
First show it is a rhombus by proving that all sides are congruent, and then show it is a rectangle by showing that one angle is a right angle.

To prove a quadrilateral is a kite:
Show that it has two pairs of consecutive sides congruent, but NOT all four sides congruent.

To prove a quadrilateral is a trapezoid:
Show it has exactly one pair of sides parallel. The other pair must be shown to not be parallel, so all four slopes need to be computed.

To prove a quadrilateral is an isosceles trapezoid:
First, show it is a trapezoid, and then show it has exactly one pair of congruent sides (the non-parallel sides).

To prove a triangle is a right triangle:
Show that one angle is a right angle.

Prove a point is on a circle:
A circle is a set of points equidistant from a given point called the center of the circle. To prove a point in on a circle, prove that the distance from the given point to the center is equal to the radius of the circle. The radius is the segment that connects the center to any point on the circle.

Examples

Example 1. Prove that quadrilateral A (1, 2), B (2, 5), C (5, 7) D (4, 4) is a parallelogram using slopes.

Solution: First, plot the points and sketch the figure. By definition, a parallelogram is a quadrilateral with two pairs of parallel sides. Verify the figure is a parallelogram by computing the slopes. The slopes of the opposite sides should be equal.

Example 2. Prove that (1, 1), (4, 4) and (6, 2) are the vertices of a right triangle.

Solution: First, plot the points and sketch the figure. The angle that appears to be the right angle is ∠B. Prove that ∠B is the right angle by determining the slopes of the two lines that make up angle B, line segments AB and BC.

Example 3. Prove that A (-3, 2), B (-2, 6), C (2, 7), D (1, 3) is a rhombus.

Solution: First, plot the points and sketch the figure. The angles do not appear to be right angles so the figure is not a square, there is no need to compute the slopes. Verify the figure is a rhombus by determining the lengths of all four sides. If they are congruent, the figure is a rhombus.

Since all sides measure √17, BA = DC = CB = DA. ABCD is a rhombus because all four sides are congruent.

Example 4. A (-4, 0), B (3, 0), and C (2, -5) represent three points of a parallelogram ABCD. Determine the coordinates of point D.

Solution: Graph the points. There are three different points that would combine with the given points to create a parallelogram, as shown in the second graph. When labeling a figure, vertices must go in order either clockwise or counter-clockwise. The only point that could be point D is (-5, -5).

Example 5. Prove or disprove that the point (1, √3) is on the circle centered at the origin and containing the point (0, 2).

Solution: First, determine the radius of the circle. The origin is the point (0, 0), the distance from the origin to (0, 2) is 2 units. If the distance from the origin to (1, √3) is also 2, then the point is on the circle. Otherwise, the point is not on the circle. Use the distance formula to determine the distance between (0, 0) and (1, √3). Squares and square roots are inverses of each other, so (√3) ² is equal to 3.

The distance is equal to the radius, so the point is on the circle.

Self-Check

# data[].attributes.attachment_link.description
Essential Question: How are the slope and the distance formula used to determine properties of polygons and circles? A figure in a coordinate plane may appear to be one of the familiar quadrilaterals, but proof is needed to show that the figure meets the definition of the particular shape.

# data[].attributes.attachment_link.readable_text
We're asked what is the equation of line B? And they tell us that line A has an equation y is equal to 2x plus 11. And they say that the line B contains the point 6, negative 7. And they tell us lines A and B are perpendicular, so that means that slope of B must be negative inverse of slope of A. So what we'll do is figure out the slope of A, then take the negative inverse of it. Then we'll know the slope of B, then we can use this point right here to fill in the gaps and figure out B's y-intercept. So what's the slope of A? This is already in slope-intercept form. The slope of A is right there, it's the 2, mx plus b. So the slope here is equal to 2. So the slope of A is 2. What is the slope of B? So what is B's slope going to have to be? Well, it's perpendicular to A, so it's going to be the negative inverse of this. The inverse of two is 1/2. The negative inverse of that is negative 1/2. So B's slope is negative 1/2. So we know that B's equation has to be y is equal to its slope, m times x plus some y-intercept. We still don't know what the y-intercept of B is, but we can use this information to figure it out. We know that y is equal to negative 7 when x is equal to 6. Negative 1/2 times 6 plus b, right? I just know that this is on the point, so this point must satisfy the equation of line B. So let's work out what b must be-- or what b, the y-intercept, this is a lowercase b, not the line B. So we have negative 7 is equal to-- what's negative 1/2 half times 6? That's not a b there, that's a 6. What's negative 1/2 times 6? It's negative 3, is equal to negative 3 plus our y-intercept. Let's add 3 to both sides of this equation, so if we add 3 to both sides-- I just want to get rid of this 3 right here-- what do we get? The left-hand side, negative 7 plus 3 is negative 4, and that's going to be equal to-- these guys cancel out-- that's equal to b, our y-intercept. So this right here is a negative 4. So the equation of line B is y is equal to-- its slope is a negative inverse of this character-- so negative 1/2, negative 1/2 x. And its y-intercept we just figured out is negative 4. And we are done.

# data[].attributes.attachment_link.title
Writing equations of perpendicular lines | Analytic geometry (video) | Khan Academy

# data[].attributes.attachment_link.description
We're asked what is the equation of line B? And they tell us that line A has an equation y is equal to 2x plus 11. And they say that the line B contains the point 6, negative 7.

# data[].attributes.body
Write equations of parallel & perpendicular lines

# data[].attributes.attachment_link.readable_text
If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# data[].attributes.attachment_link.title
Write equations of parallel & perpendicular lines | Analytic geometry (practice) | Khan Academy

# data[].attributes.attachment_link.description
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

# data[].attributes.attachment_link.title
Proof: perpendicular lines have negative reciprocal slope

# data[].attributes.attachment_link.description
Sal proves that perpendicular lines have negative reciprocal slope using triangle similarity.

# data[].attributes.attachment_link.title
Proof: parallel lines have the same slope

# data[].attributes.attachment_link.description
Sal proves that parallel lines have the same slope using triangle similarity.

# data[].attributes.body
Parallel & perpendicular lines from equation

# data[].attributes.attachment_link.readable_text
If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# data[].attributes.attachment_link.title
Parallel & perpendicular lines from equation | Analytic geometry (practice) | Khan Academy

# data[].attributes.attachment_link.description
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

# data[].attributes.attachment_link.readable_text
Essential Question: How do I compute area and perimeter of polygons in the coordinate plane?

Lesson

A polygon is a two-dimensional figure formed by three or more line segments. In this lesson, the distance formula will be applied to perimeter and area problems.

Area

The area of a polygon is the number of square units inside that polygon. The area is always measured in square units. The final area must include the appropriate unit label (feet, inches, meters, etc.). If no unit is given, use units ² or u ²

to denote square units.

Some familiar area formulas are:

Area of a rectangle = lw, length * width. The length and width of a rectangle are its sides.
Area of a parallelogram = bh, base * height. The height of a parallelogram is the perpendicular distance between two bases.
Area of a square = s ², a square is a rectangle with equal length and width.
Area of a triangle = ¹/ ₂( bh), ¹/ ₂ * base * height. The height of a triangle is the perpendicular distance from the third vertex to the base of the triangle. For a right triangle, the base and height are its legs.
Area of a trapezoid = ¹/ ₂ h( b ₁ + b ₂), where b ₁ and b ₂ are lengths of the bases. The height is the perpendicular distance between the bases.

Perimeter

The perimeter of a figure is the sum of the lengths of its sides, no matter what shape the figure is. The final answer must include the appropriate unit label (feet, inches, meters, etc.). If no unit is given, use u.

If the figure is shown in a coordinate plane, use the distance formula or the Pythagorean Theorem to determine the lengths and widths necessary before substituting the information in the appropriate formula. Sometimes the answer is not a whole number. In this case, simplify the radical and then approximate the value by using a calculator.

Examples

Example 1. Find the area of rectangle ABCD with vertices A(-3, 0), B(3, 2), C(4, -1), and D(-2, -3).

Solution: Use the distance formula to determine the length and the width of the rectangle. It is not necessary to simplify square roots.

Use the formula A = lw. It does not matter which side is the length and which is the width, although it is more common to call the longer side the length.

Example 2. Points A and B are two vertices of an equilateral triangle, what is the perimeter of the triangle? Round to two decimal places.

Solution: An equilateral triangle has 3 congruent sides. Determine the length of AB and then multiply it by 3. Use the Pythagorean Theorem or the distance formula. This example shows the Pythagorean Theorem.

P = 3(√20) ≈ 13.42 units

Example 3. Quadrilateral ABCD has vertices A (-3, 0), B (2, 4), C (3, 1), and D (-4, -3). Calculate the perimeter of the quadrilateral.

Solution: Determine the length of each side.

Minimize rounding errors by keying the entire expression into a calculator:

P = √41 + √10 + √65 + √10 ≈ 20.79 units

Example 4. Determine the area of Triangle OPQ.

Solution: The formula for the area of a triangle is A = ¹/ ₂ bh. The length of the base and the height can be determined by counting gridlines, b = 4 and h = 3, shown in pink on the diagram.

Example 5. Determine the area of Triangle ABC.

Solution: The base and height are not easily determined for this figure; they do not lie along horizontal or vertical lines. A good estimate can be found by counting the squares in the interior of the figure. There are about 6 whole squares, 13 partial squares, 1 of which is very small and will not be used in the estimate (the one by point B). Call the partial squares half squares, and the area is about 6 + 1/2(12) = 6 + 6 = 12 square units.

To actually compute the area, create a rectangle on gridlines around the triangle. The area of the triangle will be the area of the rectangle minus the area of each of the colored triangles shown. Each of these areas can be easily computed, their dimensions can be found by counting along gridlines.

Area of the rectangle = lw = 7 * 4 = 28 u ².

Area of Triangle ABC = 28 - 4.5 - 3.5 - 8 = 12 u ²

Self-Check

Self Check Problem 1. Triangle ABC has vertices A (-3, 1), B (1, 3), and C (2, -4). Calculate the perimeter of the triangle. Solution

Self Check Problem 2. Rectangle ABCD has vertices A (-3, -4), B (-1, 2), C (2, 1), and D (0, -5). Determine the area of the rectangle. Solution

Self Check Problem 3. Square PQRS has vertices P(-3, 0), Q(0, 4), R(4, 1), and S(1, -3). Determine the area of the square. Solution

Self Check Problem 4. Quadrilateral DEFG has vertices D(5, 1), E(2, 4), F(-4, 4) and G(-1, 1). Calculate the area of DEFG. Solution

Self Check Problem 5. Triangle ABC has vertices A (1, -1), B (4, 3), and C (5, -3). Calculate the area of triangle ABC. Estimate , Computation

# data[].attributes.attachment_link.title
Area and Perimeter in the Coordinate Plane

# data[].attributes.attachment_link.description
Essential Question: How do I compute area and perimeter of polygons in the coordinate plane? A polygon is a two-dimensional figure formed by three or more line segments. In this lesson, the distance formula will be applied to perimeter and area problems. The area of a polygon is the number of square units inside that polygon.

# data[].attributes.attachment_link.title
Equations of perpendicular and parallel lines

# data[].attributes.attachment_link.keywords[]
finding perimeter on coordinate plane & perimeter of irregular shapes

# data[].attributes.attachment_link.title
Finding Perimeter on Coordinate Plane & Perimeter of Irregular Shapes