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Lines, Slopes, and Perpendicularity: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

Vertex $A$ lies on $y=8$ . In triangle $ABC$ ,
$\begin{aligned}B&=(-1,1),\\C&=(5,4).\end{aligned}$
The altitude line from $A$ passes through $(2,2)$ . What is $A$ ?

Answer choices for question 1 A $(-1,8)$ B $(1,8)$ C $(2,8)$ D $(5,8)$
2

Where does the perpendicular bisector of the segment from $(-3,-1)$ to $(5,5)$ meet the line $y=-2x+8$ ?

Answer choices for question 2 A $(1,2)$ B $(-7,22)$ C $(7,-6)$ D $(3,2)$
3

Triangle $ABC$ has
$\begin{aligned}A&=(-1,0),\\B&=(5,0),\\C&=(2,6).\end{aligned}$
Where do its altitude lines from $A$ and $C$ meet?

Answer choices for question 3 A $(2,3)$ B $(2,\frac32)$ C $(0,\frac12)$ D $(\frac32,2)$
4

A line has equation $y=(k-2)x+4$ . It is perpendicular to the line through $(0,0)$ and $(3,6)$ . What is $k$ ?

Answer choices for question 4 A $\frac52$ B $0$ C $4$ D $\frac32$
5

Triangle $ABC$ has
$\begin{aligned}A&=(-3,-2),\\B&=(3,2),\\C&=(9,t).\end{aligned}$
Its median line from $C$ is parallel to $y=\frac53x-4$ . What is $t$ ?

Answer choices for question 5 A $6$ B $-15$ C $15$ D $\frac{27}5$
6

The perpendicular bisector of the segment from
$A=(-5,k)$
to
$B=(7,3)$
is vertical. Which value of $k$ and bisector equation follow?

Answer choices for question 6 A $k=-3$ ; $x=1$ B $k=3$ ; $x=1$ C $k=3$ ; $y=1$ D $k=1$ ; $x=3$
7

Consider the three lines.
$\begin{aligned}\ell:\ x+2y&=4,\\n:\ 2x+4y&=10,\\p:\ 2x-y&=1.\end{aligned}$
Which statement is correct?

Answer choices for question 7 A The first two are distinct parallels; the third is perpendicular to both; no point lies on all three B All three lines meet at one point C The first two coincide and the third is perpendicular to them D The first two are perpendicular and the third is parallel to both
8

In triangle $ABC$ ,
$\begin{aligned}A&=(-2,4),\\B&=(0,-1),\\C&=(4,1).\end{aligned}$
Find the foot of the altitude from $A$ on the full line of $BC$ .

Answer choices for question 8 A $(2,0)$ B $(-\frac25,\frac45)$ C $(0,0)$ D $(\frac25,-\frac45)$
9

A nonvertical line has slope 2. A segment has slope $-1/2$ . Which extra fact is sufficient to establish that the line is the segment's perpendicular bisector?

Answer choices for question 9 A The line contains one endpoint B The line contains the segment midpoint C The line contains a point outside the segment D The line has positive intercept
10

Two slanted lines have slopes whose product is $-1$ . Rotate the first through $90^\circ$ about a point on it. Which fact completes the converse proof that the original lines are perpendicular?

Answer choices for question 10 A The rotated line must pass through the second line at the rotation center B The original two slopes must be equal C A rotation preserves the original slope D The rotated line coincides with or is parallel to the second; perpendicularity transfers across parallels
11

Segment $AB$ has
$\begin{aligned}A&=(a,1),\\B&=(5,9).\end{aligned}$
Its perpendicular bisector has slope $1/2$ . What is $a$ ?

Answer choices for question 11 A $1$ B $21$ C $9$ D $-11$
12

Triangle $ABC$ has
$\begin{aligned}A&=(-2,-2),\\B&=(6,-2),\\C&=(2,6).\end{aligned}$
Where do the altitude line from $B$ and the median line from $C$ meet?

Answer choices for question 12 A $(2,0)$ B $(2,-2)$ C $(0,1)$ D $(6,-2)$

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