Question

lt 12.2 i can divide a directed line segment in a given ratio. /4

1. given a(3, 10) and b (6, 4). find the point p that partitions ab in a 1:2 ratio. (4 pts)
2. p(_, _

lt 12.3 i can use similar triangles to solve problems. /12

1. use the shadows to determine the height of the tall flagpole h. (4 pts)
2. h = ________
3. you plan on determining the height of a tree by using shadows. the trees shadow is 570 cm, and your shadow is 114 cm, thus creating similar triangles. determine the height of the tree if your height is 168 cm. (6 pts)
4. what theorem is used in problem 10 to determine the triangles are similar? (2 pts) 11. _______

a. sas~
b. aa~
c. sss~

Explanation:

Problem 5

Step1: Use section formula for x-coordinate

The section formula for a point dividing segment $A(x_1,y_1)$ and $B(x_2,y_2)$ in ratio $m:n$ is $x=\frac{mx_2+nx_1}{m+n}$. Here $m=1,n=2,x_1=3,x_2=6$.
$x=\frac{1\times6 + 2\times3}{1+2}$

Step2: Calculate x-coordinate

Simplify the numerator and denominator.
$x=\frac{6+6}{3}=\frac{12}{3}=4$

Step3: Use section formula for y-coordinate

Apply the section formula for y-coordinate: $y=\frac{my_2+ny_1}{m+n}$. Here $y_1=10,y_2=4$.
$y=\frac{1\times4 + 2\times10}{1+2}$

Step4: Calculate y-coordinate

Simplify the numerator and denominator.
$y=\frac{4+20}{3}=\frac{24}{3}=8$

Problem 9

Step1: Set up proportion for similar triangles

Since triangles are similar, $\frac{\text{Height of flagpole}}{\text{Its shadow}}=\frac{\text{Height of small pole}}{\text{Its shadow}}$.
$\frac{h}{8}=\frac{7}{2}$

Step2: Solve for h

Multiply both sides by 8.
$h=\frac{7\times8}{2}=28$

Problem 10

Step1: Set up proportion for similar triangles

$\frac{\text{Height of tree}}{\text{Tree's shadow}}=\frac{\text{Your height}}{\text{Your shadow}}$. Let tree height be $H$.
$\frac{H}{570}=\frac{168}{114}$

Step2: Solve for H

Multiply both sides by 570.
$H=\frac{168\times570}{114}=168\times5=840$

Problem 11

Step1: Identify similarity theorem

Both triangles have a right angle, and share the same sun angle, so two corresponding angles are equal.

Answer:

1. $P(4, 8)$
2. $h=28$ ft
3. Height of the tree = 840 cm
4. B. AA~