# Geometry of the Plane

## Self-Test:

 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.