Question

1-5. on the graph, graph line mu for m(-1, 1) and u(4, 5).
a. calculate the slope of line mu, then write an equation for the line.
b. calculate mu (the distance from m to u).
c. are there any similarities to the calculations used in parts (a) and (b)? any differences?

1-7. solve each equation, then check your solutions.
a. $\frac{x}{8} = \frac{3}{4}$
b. $\frac{2}{5} = \frac{x}{40}$
c. $\frac{1}{8} = \frac{x}{12}$
d. $\frac{x}{10} = \frac{12}{15}$

1-8. angela has a rectangular piece of paper and cuts a rectangle out of a corner as shown at right. what are the area and perimeter of the resulting shape?
diagram with 6 cm, 7 cm, 16 cm, 15 cm, 9 cm, 21 cm

Explanation:

1-5 a)

Step1: Calculate slope (m)

Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. For $M(-1,1)$ and $U(4,5)$, $x_1=-1,y_1=1,x_2=4,y_2=5$.
$m = \frac{5 - 1}{4 - (-1)} = \frac{4}{5}$

Step2: Write equation (point - slope)

Using point $M(-1,1)$: $y - y_1 = m(x - x_1)$
$y - 1 = \frac{4}{5}(x + 1)$
Simplify: $y = \frac{4}{5}x + \frac{4}{5} + 1 = \frac{4}{5}x + \frac{9}{5}$

Step1: Distance formula

Distance $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
$x_2 - x_1 = 4 - (-1) = 5$, $y_2 - y_1 = 5 - 1 = 4$.
$d = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41}$

Similarity: Both use differences of coordinates ($x_2 - x_1$, $y_2 - y_1$).
Difference: Slope uses $\frac{\Delta y}{\Delta x}$, distance uses Pythagorean theorem with those differences.

Answer:

Slope: $\frac{4}{5}$, Equation: $y = \frac{4}{5}x + \frac{9}{5}$

1-5 b)