Find the distance between the points $A(2, 3)$ and $B(5, 7)$.
Determine the distance of the point $P(-3, 4)$ from the origin $(0, 0)$.
Calculate the distance between the points $C(-1, -2)$ and $D(3, 1)$.
Which of the following correctly represents the distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$?
What is the distance between the points $E(-5, 8)$ and $F(2, 8)$?
The distance between point $A(k, -1)$ and point $B(3, 5)$ is exactly $2\sqrt{13}$. Find all possible values of $k$.
A point $P$ lies on the $x$-axis and is equidistant from the points $A(-1, 5)$ and $B(7, 1)$. Determine the exact coordinates of $P$.
Find the coordinates of the midpoint of the line segment connecting the points $A(2, 5)$ and $B(8, 1)$.
The midpoint $M$ of a line segment $PQ$ is $(4, -3)$. If the coordinates of $P$ are $(1, 5)$, what are the coordinates of point $Q$?
The line segment $PQ$ has a midpoint $M=(4, -1)$. If the coordinates of $P$ are $(2k, 5)$ and the coordinates of $Q$ are $(k+3, 3m)$, what is the value of $k+m$?
The vertices of a parallelogram $ABCD$ are $A(-2, 3)$, $B(4, 5)$, and $D(0, 7)$. Find the coordinates of the fourth vertex $C(x, y)$. (Hint: Diagonals of a parallelogram bisect each other, meaning they share the same midpoint.)
The midpoint $M$ of the line segment $AB$ is $(3, -2)$. If the coordinates of point $A$ are $(1, 4)$, what is the exact length of the segment $BM$?
Points $P(-5, 8)$, $Q(3, 4)$, and $R$ are collinear. $Q$ is the midpoint of $PR$, and $S$ is the midpoint of $QR$. Calculate the exact length of the segment $PS$.
If two non-vertical lines are parallel, what must be true about their gradients $m_1$ and $m_2$?
A line $L_1$ has the equation $y = 5x - 3$. What is the gradient of a line $L_2$ that is parallel to $L_1$?
Determine the gradient of a line that is parallel to the line defined by the equation $2x + 4y = 8$.
Which pair of linear equations represents parallel lines?
Line $L_1$ is defined by the equation $2ax - 4y = 8$. Line $L_2$ is defined by $3x + 6y = 15$. If $L_1$ and $L_2$ are parallel, determine the value of the constant $a$.
Line $A$ passes through points $P(k, 3)$ and $Q(5, 7)$. Line $B$ passes through points $R(-1, 0)$ and $S(3, 8)$. If Line $A$ is parallel to Line $B$, find the value of $k$.
A line $L$ is parallel to the line $10x + 2y = 3$. If $L$ passes through the intersection point of $y = 5x + 1$ and $y = -5x - 9$, what is the $y$-intercept of $L$?
The line $L_1$ is defined by the function $f(x) = \frac{1}{3}(6x - 9) + 2$. A second line, $L_2$, is defined by the equation $Ax + By = C$, where $A, B, C$ are constants. If $L_1$ and $L_2$ are parallel, which of the following could be the ratio $\frac{A}{B}$?
The gradient of Line A is $4$. What is the gradient of a line perpendicular to Line A?
Line M has a gradient of $-\frac{2}{3}$. What is the gradient of any line perpendicular to Line M?
Which gradient below belongs to a line perpendicular to the line described by the equation $y = \frac{5}{2}x - 1$?
Line P is a horizontal line. What is the gradient of any line perpendicular to Line P?
Line $A$ passes through the points $(1, 3)$ and $(5, k)$. Line $B$ passes through the points $(-2, 0)$ and $(4, 12)$. If Line $A$ is perpendicular to Line $B$, find the value of $k$.
Two lines are defined by the equations $3x + Ay = 5$ and $6x - 4y = 7$. If these lines are perpendicular to each other, determine the value of the constant $A$.
Line $L$ passes through the origin $O(0, 0)$ and the point $P(2a, 3b)$. Line $M$ is perpendicular to $L$ and has a gradient of $3$. Find the ratio $a:b$.
Line $L_1$ has the equation $y = (2k)x + 5$. Line $L_2$ is perpendicular to $L_1$. Line $L_3$ is parallel to $L_2$ and passes through the points $(3, 1)$ and $(-1, -7)$. Find the value of $k$.
Write the equation of the line passing through the point $(2, 5)$ with a gradient of $3$ in the gradient-point form.
Which of the following describes the gradient and a point the line $y + 1 = -2(x - 4)$ passes through?
Determine the gradient-point form of the equation for a line passing through $(-3, 1)$ with a gradient of $-\frac{1}{2}$.
If the equation $y = 4x - 7$ is written in the gradient-point form using the specific point $(2, 1)$, which of the following equations is correct?
A line $L_1$ is defined by the equation $3x - 5y = 10$. A second line, $L_2$, is perpendicular to $L_1$ and passes through the point $(-2, 4)$. Which of the following is the correct equation for $L_2$ written in the point-slope form?
Determine the equation of the line that passes through the points $P_1\left(\frac{1}{2}, -1\right)$ and $P_2(2, 5)$. If you use $P_2$ as the reference point $(x_1, y_1)$, which option represents the correct gradient-point form?
A line $L$ is parallel to the line $y = -2x + 7$. If $L$ passes through the midpoint of the segment connecting $A(1, 10)$ and $B(-5, -6)$, what is the equation of $L$ in the gradient-point form?
A linear function passes through the generic points $P_1(a, b)$ and $P_2(2a, 3b)$, where $a \neq 0$. If the equation is written in gradient-point form using $P_1$ as the fixed point, which expression is correct?
Find the equation of the line passing through the point $(-3, 5)$ with a gradient of $m = -\frac{2}{3}$. Express the answer in the form $Ax + By = C$, where $A, B, C$ are integers.
Determine the equation of the line that passes through the point $(\frac{1}{2}, -1)$ and has a gradient of $1.5$. Express your answer in the form $y = mx + c$.
A line passes through the point $(4, -1)$ and has the same gradient as the line $4x - 5y = 10$. Find the equation of this new line, expressed in the integer form $Ax + By = C$.
Determine the equation of the line passing through the point $(-\frac{3}{4}, 2)$ with a gradient $m = -4$. Express the answer in the form $Ax + By = C$, where $A, B, C$ are integers.