Points, Lines, and Planes.
What is a point?
A point is a location. It has neither shape nor
size.
Named by a capital letter
What is a midpoint?
The midpoint of a line segment is the point that divides the segment into two equal parts.
What are the four quadrants of a coordinate plane?
What is the distance between two points on a number line?
The distance between two points on a number line is the absolute value of the difference between their coordinates.
What is a line segment?
A line segment is a part of a line that has two endpoints.
What is a postulate?
A postulate is a statement that is accepted as true without proof.
How do you find the midpoint of a line segment?
Average the coordinates of the two endpoints.
What is the distance between two points on a coordinate plane?
Use the distance formula: d = J((x2 - x1) ^2 + (y2 - y1) ^2)
How do you find the distance between two points on a coordinate plane?
Use the distance formula: d = ((x2 - x1) ^2 + (y2 - y1) ^2)
How do you find the midpoint of a line segment?
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is ((x1+x2)/2, (y1+y2)[2).
How do you use postulates to prove theorems?
They are used as the building blocks for logical arguments to prove theorems.
What is a bisector?
A bisector is a line, ray, or segment that divides a geometric object into two equal parts.
What is a number line?
A number line is a visual representation of numbers on a straight line.
What is the distance formula, and how is it used?
The distance formula is d = /((x2 - x1) ^2 + (y2 - y1) ^2).
It calculates the straight-line distance between two points in a coordinate plane.
What is the slope of a line segment?
The slope of a line segment is the ratio of the change in y-coordinates to the change in x-coordinates between any two points on the line. It can be calculated as (y2 - y1) / (x2 - x1).
In geometry, why are points, line, and plane considered undefined terms?
because they are usually readily understood and are not formally explained by means of more basic words and concepts.
How do you find the equation of the perpendicular bisector of a line segment?
First, find the midpoint of the line segment. Then, find the negative reciprocal of the slope of the line segment to get the slope of the perpendicular bisector. Use the point-slope form with the midpoint and the perpendicular slope to find the equation.
How do you plot a point on a number line?
Locate the number on the line and mark a point above it.
How do you find the perimeter of a shape using the distance formula?
Calculate the distance between each pair of consecutive vertices of the shape using the distance formula. Add up all these distances to find the perimeter.
How do you find the equation of a line that passes through two given points?
Use the point-slope form of a line: y - y1 = m (x - x1), where m is the slope calculated from the two points.
Prove that the sum of the angles in a triangle is 180 degrees using postulates and theorems about parallel lines.
Draw a line parallel to one side of the triangle through the opposite vertex. Use corresponding angles and alternate interior angles to show that the angles of the triangle are congruent to the angles of a straight line, which measures 180 degrees.
Given a triangle, find the equations of the perpendicular bisectors of its sides and show that they intersect at a single point.
Find the perpendicular bisectors of each side of the triangle using the method in question 4. Solve the system of equations formed by these three lines to find the intersection point, which is the circumcenter.
Given a number line with two points, find the coordinates of the point that divides the segment into a ratio of 2:3.
Divide the distance between the endpoints into 5 equal parts. The point that divides the segment in a 2:3 ratio is 2 parts from one endpoint and 3 parts from the other.
Prove the Pythagorean Theorem using the distance formula.
Consider a right triangle with sides of length an and b and hypotenuse of length c. Place the triangle on a coordinate plane with one vertex at the origin and the other two vertices at (a, 0) and (0, b). Use the distance formula to calculate the distances between these points and show that a ^2 + b^2 = c^2.
Given two points, find the equation of the perpendicular bisector of the line segment connecting them.
First, find the midpoint of the line segment. Then, find the negative reciprocal of the slope of the line segment to get the slope of the perpendicular bisector. Use the point-slope form with the midpoint and the perpendicular slope to find the equation.