Midpoint & bisectors Unit 1
Point M is the midpoint of line segment AB. If the coordinates of A are (2,5) and the coordinates of B are (8,1), what are the coordinates of point M?
(5,3)
What is the definition of a line segment, and how does it differ from a line or a ray?
A line segment is a part of a line with two endpoints. A line extends infinitely in both directions, and a ray extends infinitely in one direction from a single endpoint.
All squares are rectangles, and shape A is a square; therefore, is shape A a rectangle?
Yes because all squares fit the definition of a rectangle.
What is the next number in the sequence: 2, 6, 12, 20, 30, —? Describe the pattern and your conjecture for the next term.
42
If it is raining, then the ground is wet.
Question: Is it true that if it is raining, the ground must be wet? What conditions might make this statement false?
not necessarily
Line segment CD intersects line segment AB at its midpoint. If the coordinates of A are (1,2) and B are (7,6), and the intersection point is M, what are the coordinates of M?
(4,4)
Explain how to find the midpoint of a line segment on a coordinate plane, given the coordinates of its endpoints.
The midpoint of a line segment with endpoints (x1,y1) and (x2,y2) is found using the formula (x1+x2/2, y1+y2/2)
Triangle PQR has vertices at P(-2,2), Q(1,6), and R(5,2). find the perimeter of triangle PQR.
12+4 squared root 2 units
If you have 1 dot, it requires O lines to connect. 2 dots needs 1 line. 3 dots needs 3 lines and 4 dots need 6 lines. How many lines are needed to connect 5 dots?Describe the pattern and provide a conjecture for the number of lines needed for n dots.
10 lines
If a student studies hard, then they will pass the exam.
Question: Does studying hard guarantee that a student will pass the exam? What other factors might influence whether they pass or fail?
No
In triangle PQR, QS is the angle bisector of angle PQR. If angle PQS is 35 degrees, what is the measure of angle PQR?
70 degrees
Describe how you can determine if two line segments are congruent.
Two line segments are congruent if they have the same length. If the lengths are equal then the segments are congruent.
If a student gets over 70% on all assignments, they will pass the course. Sarah got over 70% on all assignments, so will Sarah pass the course?
Yes because she met the condition to pass.
If today is Tuesday, what day of the week will it be 100 days from now? Explain the pattern you observed and your method for predicting the day.
Thursday
A shape is a square if and only if it has four equal sides and four right angles. Question: Can a shape be definitively identified as a square if and only if it has four equal sides and four right angles?Are there any exceptions or other shapes that might fit this description under certain conditions?
Yes
Point X is the midpoint of segment YZ. The coordinates of X are (4,-3), and the coordinates of Y is (1,2). Find the coordinates of Z.
(7,-8)
If a line segment is divided into two smaller segments, how are the lengths of the smaller segments related to the length of the original segment?
The sum of lengths of the smaller segments is equal to the length of the original segment.
Find the distance between the points A(1,5) and B(4,1) on the coordinates plane
5 units
The first few triangular numbers are 1, 3, 6, 10. What are the next two triangular numbers? Describe the pattern and state a conjecture for finding the nth triangular number.
15 and 21
You can vote in an election if and only if you are a registered citizen.
• Question: Is it universally true that voting in an election requires you to be a registered citizen, and does being a registered citizen automatically grant you the right to vote?
Are there any situations where this biconditional might not hold?
Generally yes but there would be exceptions.
If ray BD bisects angle ABC, and the measure if angle ABD is 42 degrees, what is the measure of angle DBC?
42 degrees
How do parallel line segments differ from perpendicular line segments, and what are the geometric properties associated with each?
Parallel line segments never intersect and have the same slope. Perpendicular line segments intersect at a right angle, and the product of their slopes is -1.
Every day, John wakes up before sunrise. Today is a day, so will John wake up before sunrise today?
Yes because it's a general rule that always happens.
Digital Sum Pattern: Calculate the digital sum (sum of digits) of the multiples of 9 (9, 18, 27, 36,...). What pattern do you observe in the digital sums? Make a conjecture about why this pattern occurs.
Always 9
If you eat too much sugar, then you will feel sick.
Question: Does eating too much sugar always result in feeling sick? What are some variables that could prevent this outcome?
Not always