Answer:
The required answer is [tex]c=7\sqrt{3}[/tex]
Therefore the number in green box should be 7.
Step-by-step explanation:
Given:
AB = 7√2
AD = a , BD = b , DC = c , AC = d
∠B = 45°, ∠C = 30°
To Find:
c = ?
Solution:
In Right Angle Triangle ABD Sine identity we have
[tex]\sin B = \dfrac{\textrm{side opposite to angle B}}{Hypotenuse}\\[/tex]
Substituting the values we get
[tex]\sin 45 = \dfrac{AD}{AB}= \dfrac{a}{7\sqrt{2}}[/tex]
[tex] \dfrac{1}{\sqrt{2}}= \dfrac{a}{7\sqrt{2}}\\\\\therefore a=7[/tex]
Now in Triangle ADC Tangent identity we have
[tex]\tan C = \dfrac{\textrm{side opposite to angle C}}{\textrm{side adjacent to angle C}}[/tex]
Substituting the values we get
[tex]\tan 30 = \dfrac{AD}{DC}= \dfrac{a}{c}\\\\\dfrac{1}{\sqrt{3}}=\dfrac{7}{c}\\\\\therefore c=7\sqrt{3}[/tex]
The required answer is [tex]c=7\sqrt{3}[/tex]
