Answer:
The length of AD is 4 cm
Step-by-step explanation:
* Lets explain how to solve the problem
- In Δ ABC
- Point D lies on side AB ans point E lies on side BC
- Segment DE parallel to the side AC
- AB = 16 cm , AC = 20 cm , DE = 15 cm
* Lets solve the problem
- In Δ ABC
∵ DE // AC
∴ m∠BDE = m∠BAC ⇒ corresponding angles
∴ m∠BED = m∠BCA ⇒ corresponding angles
* Lets use similarity to prove that Δ BDE similar to Δ BAC
- In triangles BDE and BAC
∵ m∠BDE = m∠BAC
∵ m∠BED = m∠BCA
∵ ∠B is a common angle in the two triangles
∴ Δ BDE ≈ Δ BAC ⇒ by AAA
∴ Their corresponding sides are proportion
∴ [tex]\frac{BD}{BA}=\frac{DE}{AC}=\frac{BE}{BC}[/tex]
∵ AB = 16 cm , AC = 20 cm , DE = 15 cm
∴ [tex]\frac{BD}{16}=\frac{15}{20}[/tex]
∴ [tex]\frac{BD}{16}=\frac{3}{4}[/tex]
- By using cross multiplication
∴ 4 BD = 16 × 3
∴ 4 BD = 48
- Divide both sides by 4
∴ BD = 12 cm
∵ Point D divides side AB into two parts BD and DA
∴ AB = BD + AD
∵ AB = 16 cm
∵ BD = 12 cm
∴ 16 = 12 + AD
- Subtract 12 from both sides
∴ AD = 4 cm
* The length of AD is 4 cm
