Question

b. find $y$. show all work.
hint (b):
how are these triangles related? how can that relationship help you find $y$?
step (b)
triangle atp: side at = 14, side ap = 18, side tp = 21
triangle odg: side od has one congruence mark matching at, side og has two congruence marks matching ap, side dg = $2y+7$ has three congruence marks matching tp

Explanation:

Step1: Identify triangle similarity

The triangles have corresponding sides marked congruent in pairs, so they are similar (SSS similarity). Corresponding sides are proportional: $\frac{14}{\text{side } OD} = \frac{18}{\text{side } OG} = \frac{21}{2y+7}$. Since the single, double, triple tick marks match, we use the proportionality of the triple tick sides and one pair of matching tick sides. We use $\frac{14}{\text{side } OD} = \frac{21}{2y+7}$, but since the triangles are similar with matching side ratios, we can directly use the ratio of the sides with single ticks to triple ticks: $\frac{14}{\text{side } OD} = \frac{21}{2y+7}$, and since side $OD$ corresponds to $TA=14$, the ratio of similarity is $\frac{14}{14}=1$? No, wait, correct proportionality: $\frac{TA}{DO} = \frac{TP}{DG}$. Since $TA=14$, $DO$ has the same tick mark so $DO=14$, $TP=21$, $DG=2y+7$. So $\frac{14}{14} = \frac{21}{2y+7}$ simplifies to $\frac{21}{2y+7}=1$.

Step2: Set up proportional equation

$$\frac{21}{2y+7} = 1$$

Step3: Solve for y

Multiply both sides by $2y+7$:
$21 = 2y + 7$
Subtract 7 from both sides:
$21 - 7 = 2y$
$14 = 2y$
Divide by 2:
$y = \frac{14}{2}$

Answer:

$y=7$