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Questions and Answers
Given points $P(3, 1)$ and $Q(-1, a)$, and that the distance between them is 5 units, find the possible values of $a$.
Given points $P(3, 1)$ and $Q(-1, a)$, and that the distance between them is 5 units, find the possible values of $a$.
$a=4$ or $a=-2$
Two lines have gradients $-\frac{2}{3}$ and $\frac{a}{6}$. If the lines are perpendicular, what is the value of $a$?
Two lines have gradients $-\frac{2}{3}$ and $\frac{a}{6}$. If the lines are perpendicular, what is the value of $a$?
$a = 9$
ABCD is a parallelogram. Given $B(2, 4)$, $C(0, 1)$, and $D(-4, 0)$, find the coordinates of $M$, the point where the diagonals meet.
ABCD is a parallelogram. Given $B(2, 4)$, $C(0, 1)$, and $D(-4, 0)$, find the coordinates of $M$, the point where the diagonals meet.
$M(-2, \frac{1}{2})$
The line through $(2, 1)$ and $(-3, n)$ has a gradient of $-4$. Find the value of $n$.
The line through $(2, 1)$ and $(-3, n)$ has a gradient of $-4$. Find the value of $n$.
Rewrite the equation $4x + 3y = 7$ in the slope-intercept form. Identify the slope and the y-intercept.
Rewrite the equation $4x + 3y = 7$ in the slope-intercept form. Identify the slope and the y-intercept.
Find the equation of the line that passes through the midpoint of the segment connecting the points $(1, 4)$ and $(3, 2)$ and has a gradient of $-2$.
Find the equation of the line that passes through the midpoint of the segment connecting the points $(1, 4)$ and $(3, 2)$ and has a gradient of $-2$.
Determine whether the point $(-2, 5)$ lies on the line defined by the equation $2y + 6x = -2$. Explain your reasoning.
Determine whether the point $(-2, 5)$ lies on the line defined by the equation $2y + 6x = -2$. Explain your reasoning.
Describe the relationship between the equations $y = 2x + 3$ and $2y = 4x + 6$. How would their graphs appear on the coordinate plane?
Describe the relationship between the equations $y = 2x + 3$ and $2y = 4x + 6$. How would their graphs appear on the coordinate plane?
Points A(-1, 4) and B(2, -3) are given. Determine the equation of the line that passes through points A and B.
Points A(-1, 4) and B(2, -3) are given. Determine the equation of the line that passes through points A and B.
Triangle PQR has vertices P(0, 1), Q(-1, -2), and R(3, -3). Classify triangle PQR based on its side lengths. Is it scalene, isosceles, or equilateral?
Triangle PQR has vertices P(0, 1), Q(-1, -2), and R(3, -3). Classify triangle PQR based on its side lengths. Is it scalene, isosceles, or equilateral?
Point A is at (2, 4) and point B is at (k, -1). If the distance between A and B is $\sqrt{29}$ units, find the possible values of k.
Point A is at (2, 4) and point B is at (k, -1). If the distance between A and B is $\sqrt{29}$ units, find the possible values of k.
A quadrilateral OABC has vertices O(0,0), A(5,0), B(8,4), and C(3,4). Given that OABC is a rhombus, demonstrate that the diagonals [OB] and [AC] are perpendicular using gradients.
A quadrilateral OABC has vertices O(0,0), A(5,0), B(8,4), and C(3,4). Given that OABC is a rhombus, demonstrate that the diagonals [OB] and [AC] are perpendicular using gradients.
Two lines have gradients $\frac{4}{-d}$ and $-\frac{d}{4}$. Find the value of $d$ if these lines are perpendicular.
Two lines have gradients $\frac{4}{-d}$ and $-\frac{d}{4}$. Find the value of $d$ if these lines are perpendicular.
The line passing through the points (a, 1) and (2, -1) has a gradient of -2. Calculate the value of a.
The line passing through the points (a, 1) and (2, -1) has a gradient of -2. Calculate the value of a.
Determine the simultaneous solution to the system of equations:
$\begin{cases}
x + 3y = 9 \
2x - y = 4
\end{cases}$
Determine the simultaneous solution to the system of equations:
$\begin{cases} x + 3y = 9 \ 2x - y = 4 \end{cases}$
The graph shows the outflow of water from a tank over time. If the graph is a straight line, what does this indicate about the rate of outflow, and what visual evidence from the graph supports this conclusion?
The graph shows the outflow of water from a tank over time. If the graph is a straight line, what does this indicate about the rate of outflow, and what visual evidence from the graph supports this conclusion?
Flashcards
Point of Intersection
Point of Intersection
A point where two or more lines intersect or cross each other.
Distance Formula
Distance Formula
The distance between two points A(x1, y1) and B(x2, y2) is √((x2 - x1)² + (y2 - y1)²).
Midpoint Formula
Midpoint Formula
The point that divides a line segment into two equal parts. Midpoint M of line segment AB is ((x1+x2)/2, (y1+y2)/2).
Gradient of a Line
Gradient of a Line
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Parallel Lines
Parallel Lines
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Perpendicular Lines
Perpendicular Lines
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Parallelogram
Parallelogram
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Y-Intercept
Y-Intercept
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What is Gradient?
What is Gradient?
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What is a Midpoint?
What is a Midpoint?
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Interpreting Gradient
Interpreting Gradient
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Scalene Triangle
Scalene Triangle
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Rhombus
Rhombus
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Study Notes
- Coordinate geometry is covered in Chapter 6
Finding the Intersection of Two Equations
- Equations must be rearranged into the form y = mx + c
- For example, 4x + 3y = 10 becomes y = (-4/3)x + 10/3
- For example, x - 2y = -3 becomes y = x/2 + 3/2
- Enter both functions in the form Y₁ = (-4/3)X + 10/3 and Y₂ = X/2 + 3/2
- Draw the graphs of both functions on the same set of axes and adjust the viewing window if necessary
- Use built-in functions to calculate the point of intersection
- In this case, the point of intersection is (1, 2).
Review Set 6A:
- For points A(3, -2) and B(1, 6):
- The distance from A to B needs to be calculated
- The midpoint of [AB] needs to be calculated
- The gradient of [AB] needs to be determined
- Coordinates of C need to be found if B is the midpoint of [AC]
- Given points P(3, 1) and Q(-1, a) are 5 units apart, the value of a needs to be determined
- Use the distance formula to classify triangle PQR given points P(3, 2), Q(-1, 4), and R(-1, 0)
- Two lines have gradients of -2/3 and a/6; if the lines are parallel or perpendicular, the value of a needs to be determined in each case
- In parallelogram ABCD:
- Coordinates of M, where the diagonals meet, need to be found
- Coordinates of point A need to be found
- The line through (2, 1) and (-3, n) has a gradient of -4; n needs to be determined
- The gradient and y-intercept of the line must be found
- y = 2/5 x + 3
- 2x - 3y = 8
- The equation of the line must be found for:
- Line (1), which is the x-axis
- Line (2)
- Line (3)
- The equation of the line needs to be found, given:
- A gradient of 5 and a y-intercept of -2
- A gradient of 2/3 passing through (3, -1)
- Passing through (4, -3) and (-1, 1)
- The graph of the line with equation x - 2y = 6 must be sketched
- Whether the point (-3, 4) lies on the line with equation 3x - y = 13 needs to be determined
- The point of intersection of 2x + y = 10 and 3x - y = 10, should be found by graphing the two lines on the same set of axes
- Given a graph indicating the number of liters of water which run from a tank over a period of time:
- Gradient of the line needs to be determined
- The meaning of the gradient in the context of water outflow needs to be interpreted
- Whether the rate of outflow of water is constant or variable needs to be determined, with supporting evidence
Review Set 6B:
- For points A(-1, 4) and B(2, -3):
- Needs to be calculated, the distance from B to A
- Needs to be calculated, the midpoint of [AB] -Needs to be found, he gradient of [AB]
- Needs to be found, the equation of the line passing through A and B
- Classify triangle PQR according to its distances, where P(0, 1), Q(-1, -2) and R(3, -3)
- A(2, 4) and B(k, -1) are √29 units apart; determine k
Additional Problems:
- Illustrates the distance traveled for different amounts of fuel at 50 kmh⁻¹ (graph A) and 80 kmh⁻¹ (graph B)
- Determine the gradient of each line
- Determine what these gradients mean
- If fuel costs $1.17 per liter, how much more would it cost to travel 1000 km at 80 kmh⁻¹ compared with 50 kmh⁻¹?
- Prove that OABC is a rhombus
- Showing using gradients, that the diagonals [OB] and [AC] are perpendicular.
- Determine d, given two lines have gradients of -4/d and d/25
- When lines are perpendicular
- When lines are parallel
- Find a, given the line through (a, 1) and (2, -1) has a gradient of -2
- Find the gradient and y-intercept of the lines with the equations:
- y = (x+2)/3
- 3x + 2y = 8
- Finding the equation of the line:
- With a gradient of 5 and a y-intercept of -1
- With x and y-intercepts of 2 and -5, respectively
- Which passes through (-1, 3) and (2, 1)
- Find k, given (2, k) lies on the line with equation 2x + 7y = 41
- Graph the lines x = 2 and 3x + 4y = -12 on the same set of axes
- Graph the equations x + 3y = 9 and 2x - y = 4 on the same set of axes
- Find simultaneous solution of:
- x + 3y = 9
- 2x - y = 4
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