Prerequisites
Which point is at (3,-2) in the following graph: https://drive.google.com/file/d/1f8Azk4L1R9KB2NzGnG6MC0V4ljStkcQm/view?usp=drive_link
P
Using your formula booklet, what is the formula for finding the midpoint M of a line segment joining two points A (x1 , y1) and B (x2 , y2) in 2D?
M ( (x1 + x2)/2 , (y1 + y2)/2)
Using your formula booklet, what is the formula for finding the distance between two points A (x1 , y1) and B (x2 , y2) in 2D?
See textbook p. 154
Using your formula booklet, what is the formula for finding the midpoint M of a line segment joining two points A (x1 , y1, z1) and B (x2 , y2, z2) in 3D?
M ( (x1 + x2)/2 , (y1 + y2)/2 , (z1 + z2)/2)
Using your formula booklet, what is the formula for finding the distance between two points A (x1 , y1) and B (x2 , y2) in 3D?
See textbook p. 154
Name the point shown in this graph: https://drive.google.com/file/d/1XUIhQ4mFzklSNlQDuf2K58vvZE3zKEdP/view?usp=drive_link
(-5,2)
Explain why the formula for the midpoint makes sense!
M ( (x1 + x2)/2 , (y1 + y2)/2)
The x and y - coordinates are the average values of the x and y - values of the two joining points of the segment.
Which other equation if equivalent to the distance formula?
The Pythagorean Theorem.
Explain why this formula for M makes sense in 3D.
Because it is the same thing as 2D but with an extra dimension.
Explain why the distance formula in 3D makes sense.
Because it is the same thing as in 2D but with an extra dimension added to it.
For
y= -5 x + 3
Find the value of y if x = 3
y = -5 (3) + 3 = -15 + 3 = -12
The points A (-3, 5) and B (0, 7) lie on the diameter of a circle. Find the coordinates of the center of the circle.
M (-1.5, 6)
See example 1 p. 151
Using an example, show that the distance formula is the same thing as using the Pythagorean theorem.
Have a look here for the derivation: https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-distance-and-midpoints/a/distance-formula
What is the midpoint between A (1, 2, 3) and B (4, -5, 6)?
M (2.5 , -1.5 , 4.5)
What is the hight of the storage box [RQ] in example 4 p. 155?
See example 4 p. 155
Solve for x:
(x-1) / 2 = 1/3
(x-1) / 2 = 1/3
(x-1) = 2/3
x = 2/3 +1
x = 5/3 (=1.6666)
Think of an example which allows you to verify that the midpoint of two diagonals of a rectangle is one and the same point.
- Add a sketch as well.
You can use the image in this link to see it: https://www.toppr.com/ask/en-np/question/20orprove-that-the-diagonals-of-a-rectangle-abcd-with-vertices-a2-1-b51-c5-6/
What is the distance between the points M and N?
See image: https://drive.google.com/file/d/1f8Azk4L1R9KB2NzGnG6MC0V4ljStkcQm/view?usp=drive_link
distance = √(22+32) = 3.61 (3SF)
Solve question b.) of example 3 p. 152 in the textbook.
M (2, 2.5 , 1)
See textbook p. 152
What is the maximum length of a garden tool that can fit in the storage box of example 4 p. 155?
See example 4 p. 155
A cuboid ABCOA1B1C1O1 has dimensions 5 x 4 x 2. The vertices of the base have the following coordinates:
A (4,0,0)
B (4,5,0)
C (0,5,0)
O (0,0,0)
--> Sketch a coordinate system in 3D and the cuboid ABCOA1B1C1O1.
--> What are the coordinates of A1 , B1 , C1 and O1 ?
A1 (4,0,2)
B1 (4,5,2)
C1 (0,5,2)
O1 (0,0,2)
Check the textbook p. 152 for an image of the cuboid.
Find the value of x if M (-3, 1.5) is the midpoint between A (x, 5) and B (-1, 10).
x = -5
See example 2 p. 151
What is the distance between point A (1, 4) and the midpoint between A and B (-3 , -4)?
First find the midpoint: M (-1 , 0)
Distance [AM] = √(42+22) = 4.47 (3SF)
Find the value of x if M (0, 1 , 2) is the midpoint between A (5, x , 3) and B (-1, 4, 10).
Then 1 = (x+4) / 2
2 = x + 4
x = -2
A tracking station lies at the origin of a coordinate system with the x-axis due east, the y-axis due north and the z-axis vertically upwards. Two aircrafts have coordinates (20, 25, 11) and (26, 31, 12) relative to the tracking station.
The radar of the tracking station has a range of 40 km. Determine whether it will be able to detect both aircrafts.
The radar will be able to detect both aircraft if they are both a distance less than 40km from the station at O = (0,0,0). Using Pythagoras’ theorem (the distance formula in 3D), we calculate the aircraft to be distances
d1= √(202 +252 +112) = 33.9, and d2 = √ (262 +312 + 122) = 42.2. Therefore, the radar will be able to detect one, but not both, of the aircrafts.