Lines,points,planes
Define a line segment and provide an example using coordinates.
A line segment is part of a line that is bounded by two endpoints. Example: The segment from A(2,3) to B(5,7)
If point B is between points A and C, and AB = 3x + 1, BC = 5x - 2, and AC = 12, find x.
3x + 1 + 5x - 2 = 12 → 8x - 1 = 12 → 8x = 13 → x = 13/8.
Derive the distance formula
d = √((x2 - x1)² + (y2 - y1)²).
If angle A and angle B are complementary and A = 2x + 10, find x if B = 3x - 20.
(2x + 10) + (3x - 20) = 90 → 5x - 10 = 90 → 5x = 100 → x = 20
What has keys but can't open locks?
Piano
Explain how two lines can be parallel in a geometric context.
Two lines are parallel if they never intersect and are equidistant from each other at all points.
If A(2, 3) and C(10, 7) with B the midpoint, find the coordinates of point B.
(6,5)
Use the midpoint formula to find the midpoint of segment connecting (3, 4) and (7, 10).
((3 + 7)/2, (4 + 10)/2) = (5, 7)
If two angles are supplementary and one measures 4x + 5 while the other measures 2x + 25, find x.
(4x + 5) + (2x + 25) = 180 → 6x + 30 = 180 → 6x = 150 → x = 25.
What can travel around the world while staying in a corner?
Stamp
If two planes intersect, what shape do they form?
They form a line.
Prove that segments AD and DB are equal if A, D, B are collinear and D is the midpoint of AB.
By definition of a midpoint, AD = DB since the midpoint divides a segment into two equal parts.
Calculate the distance between points (-1, -1) and (3, 3)
d = √((3 - (-1))² + (3 - (-1))²) = √(4² + 4²) =√32
Prove that if two angles are a linear pair, they are supplementary.
By definition, a linear pair consists of adjacent angles that sum to 180 degrees, thus they are supplementary.
The more you take, the more you leave behind. What am I?
Footsteps
If points A, B, and C are collinear and AB = 4x - 2 and AC = 6x + 10, find x if BC = 10.
Set up the equation: (4x - 2) + 10 = (6x + 10). Solve for x: 4x + 8 = 6x + 10 → 2 = 2x → x = 1.
If AB = 4x+7 and BC = 6x-3, what is the length of AC in terms of x if AC = 3x + 12?
4x+8+6x-3=3x+12
x=1
AC=15
Find the midpoint between points A(2, -3) and B(8, 5).
M = ((2 + 8)/2, (-3 + 5)/2) = (5, 1).
Identify two pairs of vertical angles given that angle 1 = 3x + 15 and angle 2 = 5x - 25. Solve for x.
x=20
angle 1 and 2=75
I’m tall when I’m young, and I’m short when I’m old. What am I?
Candle
Given points A(-3, 2), B(1, 6), and C(2, 3), determine if they are collinear and why
Calculate slopes: AB = (6-2)/(1+3) = 1, AC = (3-2)/(2+3) = 1/5. Since the slopes differ, they are not collinear.
If AB=8x^2 and BC=12x^2+11 and AC=9x^2+55 find BC
x=2
BC=59
if point A is at (2,3) and point B is at (12,17), find the point that is 3/4 the way from A to B
(9.5,13.5)
If angles A and B are vertical and A = 3x + 15, and A+B= 13x+25 find x
x=5/7
What is always in front of you but can’t be seen?
The Future