Question

In triangle ABC segment DE is parallel to the side AC . (The endpoints of segment DE lie on the sides AB and BC respectively). What is the length of AD if AB=16cm, AC=20cm, and DE=15cm;

Answer 1

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

AB = 16 cm , AC = 20 cm , DE = 15 cm

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

Answer 2

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

∴ $\frac{BD}{BA}=\frac{DE}{AC}=\frac{BE}{BC}$

∵ AB = 16 cm , AC = 20 cm , DE = 15 cm

∴ $\frac{BD}{16}=\frac{15}{20}$

∴ $\frac{BD}{16}=\frac{3}{4}$

- 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