Right Triangles and Coordinates
What is the relationship between the coordinates of a right triangle and the line going through the orgin?
The coordinates of a right triangle form the legs of a right triangle, and the slope of the line through the origin is equivalent to the ratio of the vertical leg to the horizontal leg.
What does the slope of a line tell you about the relationship between two variables?
The slope represents the rate of change between the two variables, indicating how much one variable changes for each unit change in the other variable.
How can you tell if a graph represents a proportional relationship?
A graph represents a proportional relationship if it is a straight line passing through the origin (0, 0).
What is the unit rate for a car that travels 100 miles in 4 hours?
The unit rate is 100/4 = 25 miles per hour
What is the equation of a proportional relationship where the unit rate (slope) is 4?
The equation is y = 4x
How can you use the slope of a line to determine the ratio of the legs of a right triangle?
The slope of the line is equal to the ratio of the vertical leg (rise) to the horizontal leg (run) of the right triangle.
How do similarity and dilation help us investigate the slope of lines going through the origin?
Similarity and dilation preserve the slope, so if two triangles are similar, their slopes will be the same, allowing us to use the triangle's proportions to find the slope of the line.
What is the key feature of the graph of a proportional relationship that indicates it goes through the origin?
The key feature is that the line passes through the point (0, 0) and has a constant slope.
If a recipe calls for 2 cups of flour for every 3 cups of sugar, what is the unit rate for flour to sugar?
The unit rate is 2/3, meaning for every 3 cups of sugar, 2 cups of flour are used.
Write the equation for a line going through the origin with a slope of -2.
The equation is y = -2x
If the coordinates of the points on the right triangle are (3, 4) and (0, 0), what is the slope of the line connecting them?
4-0/3-0 = 4/3
If two right triangles are similar, what does that tell you about their slopes?
If two right triangles are similar, their slopes are equal because the ratio of their corresponding legs is the same.
Given the table below, graph the proportional relationship:
x: 1, 2, 3
y: 3, 6, 9
The points (1, 3), (2, 6), and (3, 9) will form a straight line through the origin with a slope of 3.
You are buying apples at $3 per pound. How much will you pay for 5 pounds of apples?
If the unit rate is 5 and the relationship is proportional, write the equation for the line.
The equation is y = 5x
What persistent feature of right triangles is similar to the slope of a line through the origin?
The ratio of the legs of the triangle (rise/run) is constant, just like the slope of the line through the origin.
The line going through the points (1, 2) and (3, 6) is dilated. What will happen to its slope?
The slope will remain the same because dilation preserves the slope of the line.
How can you use a table to determine the unit rate of a proportional relationship?
The unit rate is the constant ratio between x and y, which can be found by dividing any y-value by the corresponding x-value.
A runner completes 5 laps in 10 minutes. What is the unit rate in laps per minute?
The unit rate is 5/10 = 0.5 laps per minute
Given the slope is 1/2, write an equation for a proportional relationship.
The equations is y = 1/2x
Given the points (2, 3) and (0, 0) form a right triangle with the x-axis, what would be the equation of the line going through the origin that represents the slope of the triangle?
The slope is 3-0/2-0 = 3/2, so the equation is y = 3/2x
What is the slope of a line going through the origin and the point (5, 7)?
The slope is 7-0/5-0 = 7/5
What would be the equation of the proportional relationship from this table?
x: 2, 4, 6
y: 10, 20, 30
The equation is y = 5x, as the unit rate (slope) is 5.
You make $15 for every 3 hours you work. What is your hourly wage?
The hourly wage is 15/3 = 15 dollars per hour
A proportional relationship has a slope of -3. Write an equation for the line that models this relationship.
The equation is y = -3x