Learn – Geometry – Coordinate geometry

Coordinate plane

Exercises

Problem set

In the coordinate plane below, several points are marked.

For each of the problems below, give the coordinates of the named point.

Problem set

For each of the following problems, draw $X$ -axis and $Y$ -axis to represent a coordinate plane. Then, mark each of the following points on a coordinate plane.

$(3,4)$
$(2,-5)$
$(-5,1)$
$(-4,-2)$
$(-5,0)$
$(0,2)$

Problem set

For each of the following pairs of points, find the distance between the points.

$(2,0), (-3,0)$
$(0,-5), (0,-2)$
$(-7,4), (3,4)$
$(4,-3), (4,-7)$
$(6,-3), (10,-3)$

Problem set

For each of the following pairs of points, find the distance between the points.

$(-5,0), (-5,-7)$
$(-9,-2), (-13,-2)$
$\left(3,\frac{1}{3}\right), \left(-10,\frac{1}{3}\right)$
$\left(-\frac{7}{3},-7\right), \left(-\frac{7}{3},-77\right)$
$\left(-\frac{1}{9},-\frac{1}{9}\right), \left(-\frac{1}{9},-\frac{1}{3}\right)$
$\left(-\frac{1}{5},-\frac{1}{6}\right), \left(-\frac{1}{15},-\frac{1}{6}\right)$

Slope of a straight line

Exercises

Problem set

Can there be two different straight lines going through the same point?
Can there be two different straight lines having the same slope?
Can there be two different straight lines having the same slope and the same y-intercept?
Can there be two different straight lines having the same slope and the same x-intercept?

Problem set

In each of the following problems, identify which of the two lines graphed in blue has bigger slope.

Problem set

A ladder leaning against a wall reaches a window that is $5$ m high. The foot of the ladder is $1$ m away from the wall. What is the slope of the ladder in its current position?
A kid goes up a vertical ladder of height $6^\prime$ to reach the top of a slide. After sliding down, the kid needs to walk $3^\prime$ to begin climbing up the ladder again. What is the slope of the slide?
A vertical post of height $20$ m is held in place by two ropes tied to the ground. Each of the ropes makes an angle with the vertical post, and the points where the ropes are tied to the ground are each $4$ m away from the foot of the vertical post. What is the slope of each rope?
A ladder is leaning against a tree with a slope of $5$ to reach a branch of the tree. If the foot of the ladder is $2$ units from the base of the tree, how high is the branch?
A ladder is used to hang an art piece $5^\prime$ high on a wall. The foot of the ladder is $1^\prime$ away from the wall. A second ladder is used to hang another art piece $12^\prime$ high on the wall, and its foot is $3^\prime$ away from the wall. Which ladder is in a steeper position?
A ladder of length $13^\prime$ is leaning against a wall to reach an art piece $12^\prime$ high. What is the slope of the ladder in its current position?
A rope of length $25^\prime$ is used to anchor a flag post of length $24^\prime$ to the ground. What is the slope of the rope?

Problem set

What is the slope of the line going through the following pairs of points?

$(1,3), (5,5)$
$(-3,2), (1,0)$
$(-3,-5), (-4,2)$
$(-1,3), (-4,-3)$
$(2,-10), (10000,-10)$
$(-10,2), (-10,10000)$

Problem set

In the following figure, a straight line is graphed in blue. Using the information shown in the figure, find the values of $KL$ and $PN$ in terms of $a$ and $b$ .

X-intercept and Y-intercept

Exercises

Problem set

Can there be two different lines having the same y-intercept?
Can there be two different lines having the same x-intercept?
Can there be two different lines going through the same two points?
Can there be two different lines having the same x-intercept and the same y-intercept?

Problem set

What is the $x$ -intercept of the line going through $(-4,3)$ and $(-2, 7)$ ?
What is the $y$ -intercept of the line going through $(-2,-3)$ and $(3, -8)$ ?
What is the $x$ -intercept of the line going through $(-2,-3)$ and $(3, -3)$ ?
What is the $y$ -intercept of the line going through $(3,-2)$ and $(-3, -2)$ ?

Equation of a straight line

Exercises

Problem set

$l$ is a line with slope $2$ and a $Y$ -intercept of $1$ . If $(5,?)$ is a point on the line $l$ , find the value of $?$ .
$p$ is a line with slope $\frac{1}{2}$ and a $Y$ -intercept of $3$ . If $(6,?)$ is a point on the line $p$ , find the value of $?$ .
$t$ is a line with slope $3$ and a $Y$ -intercept of $-2$ . If $(5,?)$ is a point on the line $t$ , find the value of $?$ .
$r$ is a line with slope $-2$ and a $Y$ -intercept of $-1$ . If $(4,?)$ is a point on the line $r$ , find the value of $?$ .
$q$ is a line with slope $2$ and a $Y$ -intercept of $-5$ . If $(-3,?)$ is a point on the line $q$ , find the value of $?$ .
$s$ is a line with slope $-4$ and a $Y$ -intercept of $-1$ . If $(-5,?)$ is a point on the line $s$ , find the value of $?$ .

Problem set

In each of the following graphs, the X-axis and the Y-axis are drawn to the same scale.

Identify which of the above graphs is represented by each of the following equations.

$y = 4x-4$
$y = \frac{1}{4}x - 1$
$y = \frac{1}{4}x$
$y = -4x - 4$
$y = -\frac{1}{4}x+1$
$y = -\frac{1}{4}x-1$
$y = -4x$
$y = -4x+6$
$4x+y = 0$
$4x-y-8 = 0$
$4x+y+8=0$
$4x+y=10$
$x+4y=8$
$x-4y=0$
$x-4y-1=0$
$x+4y+1=0$

Problem set

What is the equation of a line that has slope $2$ and that goes through point $(-1, 2)$ ?
What is the equation of a line that has slope $-\frac{1}{2}$ and that goes through point $(-3, -4)$ ?
What is the equation of a line that has slope $-\frac{7}{4}$ and that goes through point $(-4, 3)$ ?
What is the equation of a line that has slope $5$ and that goes through point $(0, -4)$ ?
What is the equation of a line that has slope $-\frac{1}{3}$ and that goes through point $(0, 5)$ ?
What is the equation of a line that has slope $-\frac{9}{2}$ and that goes through point $(0, 9)$ ?

Problem set

What is the equation of a line that goes through $(3, 5)$ and $(6, 8)$ ?
What is the equation of a line that goes through $(3, 1)$ and $(1, -3)$ ?
What is the equation of a line that goes through $(-3, 2)$ and $(-1, 1)$ ?
What is the equation of a line with y-intercept $3$ and x-intercept $1$ ?
What is the equation of a line with slope $3$ and that goes through point $(0,3)$ ?
What is the equation of a line with slope $3$ and x-intercept $3$ ?
What is the equation of a line with slope $-\frac{2}{3}$ and x-intercept $5$ ?

Problem set

Determine the slope, $x$ -intercept and $y$ -intercept for each of the following lines.

$y = -3x + 4$
$x = -3y + 4$
$3x+2y=5$
$3x+2y+6=0$
$y=-8$
$x=8$

Problem set

What is the equation of a horizontal line that is $3$ units above $X$ -axis?
What is the equation of a horizontal line that is $5$ units below $X$ -axis?
What is the equation of $X$ -axis?
What is the equation of a vertical line that is $7$ units to the right of $Y$ -axis?
What is the equation of a vertical line that is $2$ units to the left of $Y$ -axis?
What is the equation of $Y$ -axis?

Problem set

Which of the following equations represent straight lines? If an equation represents a line, identify its slope, $y$ -intercept and $x$ -intercept.

$4y = 8x + 2$
$y = 0$
$y + x = x$
$x + y = y + 2x + 3$
$y = x^2$
$y = \frac{1}{x}$
$4(y + x) = 3(y + 2)$

Intersecting and parallel lines

Exercises

Problem set

What is the equation of a line parallel to $y = 3x + 6$ that goes through the point $(2,6)$ ?
What is the equation of a line parallel to $5 = 2x-3y$ that goes through point $(3,2)$ ?
What is the equation of a line parallel to $2y + 3x = 6$ with y-intercept $5$ ?
What is the equation of a line perpendicular to $y = 5x - 100.2$ that goes through $(0,5)$ ?
What is the equation of a line perpendicular to $y = -\frac{x+2}{5}$ that goes through the origin?

Problem set

Determine the point of intersection for each of the following pairs of lines.

$y = 4x + 5$ , $y = 2x + 5$
$y = x + 5$ , $y = 2x + 1$
$y = x + 5$ , $2y = 4x + 1$
$y = 2x + 5$ , $y = 2x + 3$
$2y = 4x + 2$ , $y - 2x = 1$
$y = m_1x+ 3$ , $y = m_2x+3$ such that $m_1\ne m_2$
$x = 8$ , the line going through $(2, -2)$ and $(5, -11)$
$y = 2$ , the line going through $(2, -2)$ and $(5, -10)$
$x = 0$ , $y = 2x + 3$
$x = 0$ , $y = 2x$

Problem set

In the following figure, two parallel lines are shown in blue on a coordinate plane.
1. If $LG = 2a$ units, express $OP$ in terms of $a$ and $b$ .
2. if $LG = a+3$ units, express $AP$ in terms of $a$ and $b$ .
3. if $AP = b+1$ units, express $LO$ in terms of $a$ and $b$ .

In the following figure, $ABCD$ is a parallelogram. The points $B$ and $D$ lie on the $X$ -axis. Equation of $\overleftrightarrow{AB}$ is $3x-2y-6=0$ and equation of $\overleftrightarrow{AD}$ is $x+2y+6=0$ . What are the equations of $\overleftrightarrow{BC}$ and $\overleftrightarrow{CD}$ ?

$PQRS$ is a parallelogram. The points $Q,R,S$ are $\left(-\frac{9}{2}, 2\right), (-3, 3), \left(\frac{3}{2}, -\frac{9}{2}\right)$ respectively. What is the point $P$ ?

Problem set

In the following figure, two parallel lines are graphed in blue.

The following questions are based on the above figure.

If $OA = a\mbox{ units}, OB = b\mbox{ units}$ and $OP = \frac{b}{2}\mbox{ units}$ , what is the value of $OQ$ ?
If $OA = a\mbox{ units}, OB = b\mbox{ units}$ and $OP = a\mbox{ units}$ , what is the value of $OQ$ ?
If $A = (a,0), B = (0,b)$ and $OP = \frac{b}{2}\mbox{ units}$ , what is the point $Q$ ?
If $OA = a\mbox{ units}, B = (0,b)$ and $OP = \frac{b}{2}$ , what is the equation of the line $\overleftrightarrow{PQ}$ ?
If $OQ = a\mbox{ units}, OA = 2a\mbox{ units}$ and $P = (0,b)$ , what is the equation of line $\overleftrightarrow{PQ}$ ?

Distance between points

Exercises

Problem set

What is the distance between the following pairs of points?

$(2,2), (5,6)$
$(10,4), (2,-2)$
$(0,0), (-2,-2)$
$(-7,-7), (-6,-6)$
$(-7,2), (-2,-10)$

Problem set

What is the distance between the following pairs of points?

What is the formula for distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ ?
Does $y = 3$ represent a line? What is its y-intercept? Find points on the line that are at a distance of $3$ units from its y-intercept.
Does $x = 3$ represent a line? Find equations of the two lines parallel to this line at a perpendicular distance of $5$ units from it.
Does the equation $3x = -4y$ represent a line? If so, write in standard form. What is its y-intercept? Observe that this line passes through the origin. Find points on the line that are $5$ units away from the origin.

Miscellaneous

Exercises

Problem set

$(-2, -3), (3, 7)$ are two diagonal end points of a rectangle. One of the sides of the rectangle is horizontal. Find all the vertices of the rectangle.
$(-2, -3), (0, 0)$ are two diagonal end points of a rectangle. Two of the vertices of the rectangle lie on the Y-axis. Find the area of the rectangle.
$(-1, 3), (4, 3), (2, b)$ are the three vertices of a triangle. The area of the triangle is $15$ square units. What are the possible values of $b$ ?
$(3, -10), (3, -6), (a, 10)$ are the three vertices of a triangle. The area of the triangle is $10$ square units. What are the possible values of $a$ ?

Problem set

Are $(4,3), (1,-3), (5,5)$ colinear?
Are $(-9,3), (-10,3), (-9,4), (-10,4)$ colinear?
Are $(5, 3), (5, 4), (5, -100)$ colinear?
$(-1, -2), (0,0), (6,b)$ are colinear. What is the value of $b$ ?
$(-1, -2), (b,0), (3b,2)$ are colinear. What is the value of $b$ ?

Problem set

Which of the following points lie on the graph of the equation $y = 3x^3-3x^2-24x+14$ ?

$(-1,12)$
$(1,10)$
$(-3,-22)$
$(3,-4)$
$(-2,26)$

Problem set

A circle of radius $1$ unit has origin as its center. Identify its $x$ -intercepts and $y$ -intercepts.
What is the equation of the line that is perpendicular to $3x-4y+100=0$ and that goes through the point $(-3,5)$ ?
What is the y-intercept of the line with slope $3$ and x-intercept $1$ ?
A line goes through the points $(-1, 6)$ and $(2, -3)$ . Find a point on the line that is to the left of the vertical line $x = -2.5$ .
A line with slope $-2$ and $x$ -intercept $1$ is translated horizontally to the left by $2$ units. What is the $y$ -intercept of the new line generated?