Answer:
DE = 8
Step-by-step explanation:
ΔABC and ΔDBE are similar.
As they share the same angles, the sides are in proportion to each other.
Therefore, we can solve using ratios.
Tan trigonometric ratio
[tex]\sf \tan(\theta)=\dfrac{O}{A}[/tex]
where:
- [tex]\theta[/tex] is the angle
- O is the side opposite the angle
- A is the side adjacent the angle
Given:
[tex]\sf \tan (B)=\dfrac{4}{3}[/tex]
Therefore:
[tex]\implies \sf \dfrac{DE}{BD}=\dfrac{AC}{BA}=\dfrac{4x}{3x}[/tex]
We know that BC = 15 and that AC : BA = 4x : 3x
Using Pythagoras' Theorem:
[tex]\implies \sf (3x)^2+(4x)^2=15^2[/tex]
[tex]\implies \sf 9x^2+16x^2=225[/tex]
[tex]\implies \sf x^2(9+16)=225[/tex]
[tex]\implies \sf x^2=9[/tex]
[tex]\implies \sf x=3[/tex]
Therefore,
⇒ AC = 4 · 3 = 12
⇒ BA = 3 · 3 = 9
Now we have found the length of BA, we can find the length of BD:
⇒ BD = BA - DA
⇒ BD = 9 - 3
⇒ BD = 6
As DE : BD = 4x : 3x and BD = 6 then:
⇒ 3x = 6
⇒ x = 2
Therefore, to find DE, substitute x = 2:
⇒ DE = 4x
⇒ DE = 4(2)
⇒ DE = 8
