(1) AC is 70 units long.That is, AS + RS + RC = 70
We need to know the value of RS in order to find the area of the square. But the above equation also has 2 other unknowns. Even if we try to express AS and RC in terms of RS, we cannot do so without involving other dimensions of the triangles in this figure. Therefore, St. 1 is not sufficient to determine a unique value of RS.
(2) The product of the length of AS and the length of RC is 396.That is,
AS*RC = 396In right triangle ABC,
\(tanC = \frac{AB}{BC}\) . . . (1)
In right triangle CRQ,
\(tanC = \frac{QR}{RC}\) . . . (2)
By equating (1) and (2), we get:
\(\frac{AB}{BC} = \frac{QR}{RC}\)
That is,
\(RC = \frac{QR*BC}{AB}\) . . . (3)
Now, in right triangle ABC,
\(tanA = \frac{BC}{AB}\) . . . (1')
In right triangle ASP,
\(tanA = \frac{PS}{AS}\) . . . (2')
By equating (1') and (2'), we get:
\(\frac{BC}{AB} = \frac{PS}{AS}\)
That is,
\(AS = \frac{PS*AB}{BC}\) . . . (3')
By substituting equations (3) and (3') in the red equation above, we get:
\(\frac{PS*AB}{BC}*\frac{QR*BC}{AB} = 396\)
That is, PS*QR = 396
\((Side of square)^2 = 396\)
So, the area of square = 396 sq. units.
Thus, St. 2 is sufficient to find the area of the square.
Hope this helped!
Best Regards
Japinder
Thanks for this detailed explanation.
1 question though, do we need to know trigonometry to this level for the GMAT. I would like a Q51
Kudos if you like my response please