Geometry of the Plane

1)       The equation of a line with slope $4$ that goes through the point $( 3, 2 )$ is:

$y = 5x + 4$

$y = (-2/3)x + 4$

$y = (-1/2)x + 4$

$y = 4x - 5$

$y = 4x - 10$

Hint A formula for a line is: $$ y-y_1 = m(x-x_1) $$ where $m$ is the slope and $(x_1,y_1)$ is a point on the line.

2)       Which pair of lines are perpendicular to each other?
$\begin{array}{rl} y & = x+2 \\ y & = -x+2 \end{array}$	$\begin{array}{rl} y & = 3x-1 \\ y & = \frac13x+2 \end{array}$
$\begin{array}{rl} y & = -\frac23x-1 \\ y & = -\frac32x+5 \end{array}$	$\begin{array}{rl} y & = -3x+4 \\ y & = -3x-\frac14 \end{array}$
$\begin{array}{rl} y & = -3x+5 \\ y & = 5x-3 \end{array}$

Hint Two lines are perpendicular if the slopes are the negative reciprocal of one another.

3)       The distance between   $( 5, -3 )$   and   $( 2, 1 )$ is

less than $2$

between $2$ and $4$ inclusive

strictly between $4$ and $5$

between $5$ and $7$ inclusive

strictly greater than $7$

Hint The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is: $$ \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2} $$

4)       The graph of the circle of radius $14$, centered at $( 6, 2 )$ includes:

the point   $( 18, 4 )$

the point   $( -8, 2 )$

the point   $( 0, 10 )$

the point   $( 1, 11 )$

not any of the above points

Hint A formula for a circle is: $$ (x-x_C)^2 + (y-y_C)^2 = r^2 $$ where $r$ is the radius and $(x_C,y_C)$ the centre.

5)       If a parabola with vertex $( 2, 5 )$ passes through the point $( 1, 8 )$ and has an equation of the form $y=a(x-h)^2+k$,   then $a=$

$1$

$-3$

$3$

$\frac19$

$\frac38$

Hint 1 The parabola with this equation passes through the point $(h,k)$.
Hint 2 After the first hint, plug in the values for the second point to obtain an equation for $a$.

6)       Two line segments $AB$ and $BC$ meet at a right angle at point $B$. Their lengths are $6$ and $8$ respectively. The distance from points $A$ to $C$ is

$4$

$14$

$22$

$100$

$10$

Hint Make a drawing.
Use the Pythagorean Theorem.

7)       Let $A$ be the point   $(1,2)$   and $B$ be the point   $(5,-1)$.   The equation of the line that is perpendicular to $AB$ and passes through the midpoint of $AB$ is:

$y=-\frac34 x - \frac{17}{4}$

$y=\frac43x-\frac72$

$y=\frac34x-\frac74$

$y=-\frac23x-\frac52$

$y=-\frac32x+5$

Hint 1 Compute the slope of the line.
Compute the midpoint.
Hint 2 A formula for a line is: $$ y-y_1=m(x-x_1) $$ where $m$ is the slope and $(x_1,y_1)$ is a point on the line.

8)       The line $\ell$ intersects the $y$-axis at $y = 5$ and the $x$-axis at $x = 3$. The line $\ell$ is parallel to:

$5x + 3y = -4$

$5x - 3y = 4$

$-5x + 3y = 4$

$5y + 3x = -4$

$5y - 3x = 4$

Hint 1 A formula for the slope of a line is: $$ m = \frac{y_1-y_2}{x_1-x_2}. $$
Hint 2 Two lines are parallel if the slopes are equal.

9)       If you reflect the graph of $y = 3x + 2$ about the $y$-axis, then:

no points will be fixed.

the point   $( -1, -1 )$   will be fixed.

the point   $(-\frac23, 0)$   will be fixed.

the point   $( 0, 2 )$   will be fixed.

more than one point will be fixed.

Hint The graph of $y=f(x)$, when reflected on the $y$-axis, becomes $y=f(-x)$.

10)       The quadrilateral with vertices $( -2, -1 )$,   $( -2, -5 )$,   $( 5, 0 )$,   and   $( 5, 4 )$ is :

a trapezoid

a parallelogram that is not a rhombus

a rhombus

a rectangle that is not a square

a square

Hint Make a drawing.
Check the slopes of the sides of the quadrilateral.