what is c , the length of the squares's side.
Answers
The figure is also showed in the attachment 1 including some changes.
Let the line segments of lengths 12, 3 and 9 units in the square be named as AB, BC and CD each respectively, as in the attachment 1.
Now,
⇒ Extend AB towards top right. (Attachment 2)
⇒ Draw a line perpendicular to CD at D. (Attachment 3)
⇒ Mark the point of intersection of this perpendicular bisector and AB produced as E. (Attachment 3)
⇒ Join AD. (Attachment 4)
Here, AD is one diagonal of the square. We have to find the length of AD first to get the value of x.
BCDE is a rectangle, so that BC = DE = 3 units and BE = CD = 9 units.
Consider ΔADE. It is a right triangle.
Thus, according to Pythagoras' theorem,
AD² = AE² + DE²
⇒ AD² = (AB + BE)² + DE²
⇒ AD² = (12 + 9)² + 3²
⇒ AD² = 21² + 3²
⇒ AD² = 441 + 9
⇒ AD² = 450
⇒ AD = √450
⇒ AD = √225 × 2
⇒ AD = 15√2 units
Thus we got the length of AD.
In any square, the length of a diagonal is √2 times that of a side.
Here the length of side of the square is x. So the length of the diagonal will be x√2 in terms of x, which is equal to 15√2 units.
Thus,
x√2 = 15√2
⇒ x = 15
Hence the value of x is 15 units.
Another method...!!!
Mark the point of intersection of BC and AD as F. (Attachment 4)
Consider triangles ABF and DCF.
∠ABF = ∠DCF = 90°
∠AFB = ∠DFC (alternate)
∴ ΔABF ~ ΔDCF
As they are similar, the sides corresponding to equal angles are proportional.
∴ AF : DF = BF : CF = AB : CD = 12 : 9 = 4 : 3 → (1)
∴ BF : CF = 4 : 3
BF = BC × 4 / (4 + 3)
⇒ BF = 3 × 4 / 7
⇒ BF = 12/7 units
Consider ΔABF.
According to Pythagoras' theorem,
AF² = AB² + BF²
⇒ AF² = 12² + (12/7)²
⇒ AF² = 144 + (144/49)
⇒ AF² = (7056 + 144)/49
⇒ AF² = 7200/49
⇒ AF = √(7200/49)
⇒ AF = (60√2)/7 units
From (1),
AF : DF = 4 : 3
AF = AD × 4/(4 + 3)
⇒ (60√2)/7 = AD × 4/7
⇒ (60√2)/7 = 4AD/7
⇒ 60√2 = 4AD
⇒ AD = (60√2)/4
⇒ AD = 15√2 units
Here we also get the length of AD as 15√2 units. So we can find the value of x as we found earlier.
According to the concept that 'the length of the diagonal of a square is √2 times that of side',
x√2 = 15√2
x = 15 units
