Substitute the given measurements DA = 4 cm, DE = 12 cm, and DW = 24 cm into the ratio. Solving for DB, we find DB = 8 cm.

In PQR, since ST QR, we apply the Basic Proportionality Theorem to relate the segments: (PS)/(SQ) = (PT)/(TR). Since TR = PR - PT, we can express the entire right-hand side in terms of PT and PR.

Example 13. In $\triangle DEW$ , $AB \parallel EW$ . If $AD = 4\text{ cm}$ , $DE = 12\text{ cm}$ and $DW = 24\text{ cm}$ , find the length of $DB$ . Figure Description: Triangle $DEW$ with vertex $D$ at the top, base $EW$ at the bottom, and a line segment $AB$ parallel to $EW$ intersecting side $DE$ at point $A$ and side $DW$ at point $B$ . Example 14. In $\triangle PQR$ , $ST \parallel QR$ , $\frac{PS}{SQ} = \frac{3}{5}$ and $PR = 28\text{ cm}$ , find the length of $PT$ . Figure Description: Triangle $PQR$ with vertex $P$ at the top, base $QR$ at the bottom, and a line segment $ST$ parallel to $QR$ intersecting side $PQ$ at point $S$ and side $PR$ at point $T$ .

Answer: For Example 13, $DB = 8\text{ cm}$ . For Example 14, $PT = 10.5\text{ cm}$ .

Step-by-step solution

Step 1: State Basic Proportionality Theorem for Example 13

In $\triangle DEW$ , since segment $AB \parallel EW$ , by the Basic Proportionality Theorem (Thales's theorem), the line dividing the sides creates proportional segments, giving $\frac{DA}{DE} = \frac{DB}{DW}$ .

Step 2: Calculate DB in Example 13

Substitute the given measurements $DA = 4\text{ cm}$ , $DE = 12\text{ cm}$ , and $DW = 24\text{ cm}$ into the ratio. Solving for $DB$ , we find $DB = 8\text{ cm}$ .

Step 3: State Proportionality relation for Example 14

In $\triangle PQR$ , since $ST \parallel QR$ , we apply the Basic Proportionality Theorem to relate the segments: $\frac{PS}{SQ} = \frac{PT}{TR}$ . Since $TR = PR - PT$ , we can express the entire right-hand side in terms of $PT$ and $PR$ .

Step 4: Substitute values and solve for PT in Example 14

Substitute $\frac{PS}{SQ} = \frac{3}{5}$ and $PR = 28\text{ cm}$ . Expanding the cross-multiplication gives $84 - 3PT = 5PT$ , which simplifies to $8PT = 84$ , giving $PT = 10.5\text{ cm}$ .

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