Answer:
The Value of [tex]a=\frac{15}{4}[/tex].
Step-by-step explanation:
We have Named the figure please find the attachment for your reference.
Given:
PR = y
QR = a
RS = b
PS = z
PQ = x
QS = 15
∠P = 90°
∠R = 90°
∠Q = 60°
∠S = 30°
We need to find the Value of 'a'.
Solution:
Now we know that:
In Δ PQS
∠P = 90°
∠S = 30°
Now we know that;
[tex]sin\ \theta = \frac{opposite\ side}{Hypotenuse}[/tex]
[tex]sin \ S= \frac{PQ}{QS}[/tex]
Substituting the given values we get;
[tex]sin\ 30\°=\frac{x}{15}[/tex]
Now we know that;
[tex]sin\ 30\° = \frac12[/tex]
So we can say that;
[tex]\frac{1}{2}=\frac{x}{15}\\\\x=\frac{15}{2}[/tex]
Now In Triangle PQR.
∠R = 90°
∠Q = 60°
So we can say that;
[tex]Cos \theta = \frac{adjacent \ Side}{Hypotenuse}\\[/tex]
[tex]Cos\ Q = \frac{QR}{PQ}[/tex]
Substituting the given values we get;
[tex]cos 60\°= \frac{a}{x}[/tex]
Now we know that;
[tex]cos 60\°= \frac12[/tex]
[tex]x=\frac{15}{2}[/tex]
So substituting the values we get;
[tex]\frac{1}{2}=\frac{a}{\frac{15}{2}}[/tex]
By Using Cross Multiplication we get;
[tex]a= \frac{1}{2}\times\frac{15}{2}\\\\a=\frac{15}{4}[/tex]
Hence The Value of [tex]a=\frac{15}{4}[/tex].
