Notes
Distance and midpoint
- Distance: $\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
- Midpoint: $\displaystyle \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. To find a missing endpoint use $2 \cdot \text{midpoint} - \text{known endpoint}$.
Slopes of lines
- Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals (product $-1$).
Perimeter and area
- Use the distance formula for slanted sides, then apply perimeter and area formulas. A triangle's area is $\displaystyle \frac{1}{2}bh$.
Constructions
- Compass and straightedge constructions: copy a segment, perpendicular bisector, angle bisector, equilateral triangle, parallel and perpendicular lines. They rely on arcs with equal radii.
Topics in this unit
Each topic has its own short notes and practice.
Practice
The differences are $-8$ and $15$, so $\displaystyle \sqrt{64 + 225} = \sqrt{289} = 17$.
$\displaystyle \frac{-6 + 2}{2} = -2$ and $\displaystyle \frac{9 + (-1)}{2} = 4$.
The given slope is $\displaystyle \frac{5 - 1}{2 - 0} = 2$. The negative reciprocal is $\displaystyle -\frac{1}{2}$.
The legs are 6 and 8, and the hypotenuse is $\displaystyle \sqrt{36 + 64} = 10$. So $6 + 8 + 10 = 24$.
$AB$ is horizontal and $BC$ is vertical, so they are perpendicular. Also $AB = 4$, $BC = 3$ and $AC = 5$.
It cuts $\overline{AB}$ into two equal parts at a right angle.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Triangle $ABC$ has vertices $A(1, 2)$, $B(7, 2)$ and $C(7, 10)$.
(a) Use the distance formula to find the length of $\overline{AC}$.
Show the answer and scoring guide
$\displaystyle AC = \sqrt{(7 - 1)^2 + (10 - 2)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.
Check off the points you earned:
(b) Find the midpoint of $\overline{AC}$ and the slope of a line perpendicular to $\overline{AC}$.
Show the answer and scoring guide
The midpoint is $\displaystyle \left(\frac{1 + 7}{2}, \frac{2 + 10}{2}\right) = (4, 6)$. The slope of $AC$ is $\displaystyle \frac{8}{6} = \frac{4}{3}$, so a perpendicular line has slope $\displaystyle -\frac{3}{4}$.
Check off the points you earned:
(c) Find the perimeter and the area of the triangle.
Show the answer and scoring guide
$AB = 6$ and $BC = 8$, so the perimeter is $6 + 8 + 10 = 24$. The area is $\displaystyle \frac{1}{2}(6)(8) = 24$ square units.
Check off the points you earned: