Question

if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides __________.
a proportionally
in half
into two congruent parts
equally

1. find x in the figure below given that \\(\frac{am}{mb} = \frac{cn}{nb}\\).
2. solve for x.

Explanation:

Step1: Answer triangle proportionality theorem

This is the basic statement of the Triangle Proportionality Theorem (also called Thales' Theorem).

Step2: Substitute values into proportion

From the figure, $AM=18$, $MB=12$, $CN=3x$, $NB=16$. Substitute into $\frac{AM}{MB}=\frac{CN}{NB}$:
$\frac{18}{12}=\frac{3x}{16}$

Step3: Simplify left side of equation

Simplify $\frac{18}{12}$ to $\frac{3}{2}$:
$\frac{3}{2}=\frac{3x}{16}$

Step4: Solve for x

Cross-multiply: $3\times16=2\times3x$ → $48=6x$ → $x=\frac{48}{6}=8$

Step5: Set up proportion for third problem

By the Triangle Proportionality Theorem:
$\frac{x}{x+5}=\frac{x-2}{x+1}$

Step6: Cross-multiply to solve for x

$x(x+1)=(x-2)(x+5)$
Expand both sides: $x^2+x=x^2+5x-2x-10$
Simplify: $x^2+x=x^2+3x-10$
Subtract $x^2$ from both sides: $x=3x-10$
Rearrange: $10=2x$ → $x=5$

Answer:

1. A. proportionally
2. D. 8
3. A. 5