Geometry Coordinate Geometry and Proofs Lesson (Free PDF)

Geometry math lesson: Coordinate Geometry and Proofs – Maths PDF

Cover of the book Geometry Math Lessons (PDF)

Download as a PDFThe Geometry math lessons bookPrintable PDFSee the book ›

Sommaire

This coordinate geometry lesson for Geometry covers coordinate geometry and proofs (Common Core G-GPE.B.4, G-GPE.B.5, G-GPE.B.6, G-GPE.B.7), with a free PDF to print.

In the coordinate plane, parallel lines have equal slopes and perpendicular lines have slopes that are opposite reciprocals. A point that divides a segment in the ratio a to b is a/(a + b) of the way from the first endpoint. The distance formula gives side lengths, perimeters and areas, and slope and distance tests prove properties of polygons.

Geometry · Common Core State Standards: G-GPE.B.4, G-GPE.B.5, G-GPE.B.6, G-GPE.B.7

Key facts in this coordinate geometry lesson

Key facts

Parallel lines: equal slopes. Perpendicular lines: slopes multiply to -1.
Point P with AP:PB = a:b is a/(a+b) of the way from A to B.
Distance: square root of (x2 – x1) squared plus (y2 – y1) squared.
Perimeter = sum of distances; area uses base times height or decomposition.
Coordinate proofs test slopes (parallel, perpendicular) and distances (equal sides).

How do slopes show parallel and perpendicular lines?

The slope of the line through \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is \(m = \frac{y_{2} – y_{1}}{x_{2} – x_{1}}\). In the coordinate plane, two nonvertical lines are parallel when their slopes are equal, and perpendicular when the product of their slopes is \(-1\), so the slopes are opposite reciprocals.

Parallel lines have the same slope.

Perpendicular slopes are opposite reciprocals.

How do you write the equation of a parallel or perpendicular line?

Find the slope of the given line.
Use the same slope for a parallel line, or the opposite reciprocal for a perpendicular line.
Substitute the given point into \(y – y_{1} = m(x – x_{1})\) and simplify.

How do you divide a segment in a given ratio?

To find the point P that splits segment AB so that \(AP : PB = a : b\), move the fraction \(\frac{a}{a + b}\) of the way from A to B:

\[P = \left( x_{1} + \frac{a}{a + b}(x_{2} – x_{1}),\ y_{1} + \frac{a}{a + b}(y_{2} – y_{1}) \right)\]

P splits AB in the ratio 2 : 3 from A.

How do you find lengths, perimeter and area from coordinates?

The distance formula comes from the Pythagorean theorem: \(d = \sqrt{(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}}\). Add the side lengths for the perimeter. For area, use a base and height when a side is horizontal or vertical, or split the polygon into rectangles and triangles.

Slope of AB = 1/2; slope of AC = 7/4.

How do you prove a fact with coordinates?

A coordinate proof places a figure on the plane and uses slope, distance and midpoint to test a property. To show that a quadrilateral is a parallelogram, show both pairs of opposite sides have equal slopes. To show a right angle, show two sides have slopes with product \(-1\). To show a triangle is isosceles, show two side lengths are equal.

Example: The point that is \(\frac{2}{5}\) of the way from \(A(0, 0)\) to \(B(10, 5)\) is \((0 + 4,\ 0 + 2) = (4, 2)\).

Coordinate geometry lesson: questions and answers

How do you know if two lines are perpendicular?

Multiply their slopes. If the product is -1, the lines are perpendicular. Slopes 2/3 and -3/2 multiply to -1.

How do you find a point that divides a segment in a ratio?

Find the fraction a/(a + b), then add that fraction of the x-change and of the y-change to the starting point.

What is the distance formula?

The distance between (x1, y1) and (x2, y2) is the square root of (x2 – x1) squared plus (y2 – y1) squared. It comes from the Pythagorean theorem.

How can you prove a quadrilateral is a parallelogram with coordinates?

Show that both pairs of opposite sides have equal slopes, or that the diagonals have the same midpoint.

How do you find the area of a triangle on the coordinate plane?

Use one half base times height when one side is horizontal or vertical. Otherwise enclose the triangle in a rectangle and subtract the extra triangles.

How to use this coordinate geometry lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the coordinate geometry lesson as a PDF with the button below and keep it with your notes.

All Geometry chapters: Geometry math.

Voter.. post

Free printable PDF