Plot Devices (Tools of Coordinate Geometry)
What is the formula for finding the midpoint of a line segment with endpoints (x1, y1) and (x2, y2)?
M = ((x1 + x2)/2, (y1 + y2)/2)
What are the three classifications of triangles based on the lengths of their sides?
Equilateral, isosceles, or scalene
What is the standard equation of a circle with its centre at the origin and a radius 'r'?
x2 + y2 = r2
Name the theorem that states connecting the midpoints of each side of any quadrilateral results in a parallelogram.
Varignon's Parallelogram Theorem
On a map where each unit represents 1 km, what is the distance between two department stores located at (4, 3) and (1, 7)?
5 km
How can you determine if two lines are perpendicular by examining their slopes?
Their slopes are negative reciprocals of each other
If you connect the midpoints of each side of any quadrilateral, what specific type of shape will you form?
A parallelogram
What is the radius of a circle given the equation x2 + y2 = 100?
10
What theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length?
The Side-Splitting Theorem
A forest fire is at the origin (0,0). Town A is at (26,77) and Town B is at (12,–88). Which town should be the priority for evacuation?
Town A
What is the distance of the point (5, 12) from the origin?
13.0 units
In a triangle, what is a line segment that joins a vertex to the midpoint of the opposite side called?
A median
A circle has its centre at the origin and passes through the point (6, 8). What is the equation of this circle?
x2 + y2 = 100
According to the Perpendicular Bisector Theorem, what unique property does any point on the perpendicular bisector of a line segment possess in relation to the segment's endpoints?
It is the same distance from both ends of the segment
The endpoints of the diameter of a circle are at A(–1,1) and B(0.5, –3). Where is the center of the circle?
At (–0.25, –1)
Describe the three main steps for finding the shortest distance from a given point to a given line.
1. Determine the equation of the line perpendicular to the given line and passing through the point.
2. Find the intersection point between the two lines.
3. Use the distance formula between the given point and the intersection point
Name the point where the altitudes of a triangle meet, and state whether this point is inside or outside an obtuse triangle.
The orthocentre, which is outside an obtuse triangle
The Chord Perpendicular Bisector Theorem states that the perpendicular bisector of a chord within a circle always passes through what specific point?
The centre of the circle
Which theorem states that the perpendicular bisector of a chord within a circle always goes through the centre of the circle?
The Chord Perpendicular Bisector Theorem
An iceberg is at (40,10) and a cruise ship travels in a straight line from (–20,60) to (50,–10). How close will the ship get to the iceberg?
7.07 units
One endpoint of a line segment is at (–3, –1), and its midpoint is at (1, 1). What are the coordinates of the second endpoint?
(5, 3)
How can you determine if a triangle formed by three given points is a right triangle?
By using the distance formula to find the lengths of all three sides and then checking if the Pythagorean theorem holds true.
For the circle defined by x2 + y2 = 45, is the point (3, -5) located inside, outside, or on the circle?
Inside
List at least three of the coordinate geometry tools typically used to algebraically prove geometric theorems.
Equation of a line, slope formula (including parallel/perpendicular lines), distance formula, and midpoint formula
Three homes A(–16,32), B(22,–24), and C(56,8) are connecting to an electrical grid line that runs from G(–16,–30) to H(38,42). What is the total length of electrical line needed?
98.8 units