Show that the quadrilateral, formed by joining the mid – points of the sides of a square is also a square.

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70 °

80 °

90 °

50 °

answer is C.

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Detailed Solution

Before solving this question we need to know about squares and their properties.
Square: A square is a two-dimensional shape. It is a quadrilateral, which means it is made up of four straight lines. Here are some properties of a square.
1. All its sides are of equal length, in other words the length of its four sides is always the same or the same.
2. All its angles are 90°.
3. Its diagonals bisect each other.
4. Its diagonals are also 90° to each other.

Since the points P, Q, R, S divide the segments CD, BC, AB, DA equally, their lengths will be equal.
Since ABCD is a square, all its angles will be right angles.
And as we know, a straight line subtends an angle of 180°.
Therefore, the angle at "R" will also be 180°.

Now, as we can see, an isosceles triangle has been created. Here we will use the fact that the angles opposite to the same sides are also equal and the sum of the angles in a triangle is 180°. From triangle ARS we have

and

∠ A R S + ∠ S R A + ∠ R A S = 180 o ⇒ ∠ A R S + ∠ A R S + 90 o = 180 o ⇒ 2 ∠ A R S = 180 o − 90 o ⇒ 2 ∠ A R S = 90 o ⇒ ∠ A R S = 90 o 2 ⇒ ∠ A R S = 45 °

∠ A R S = ∠ S R A = 45 o

Similarly,

∠ Q R B = 45 o

∠ S R A + ∠ S R Q + ∠ Q R B = 180 o ⇒ 45 ° + ∠ S R Q + 45 o = 180 o ⇒ ∠ S R Q = 180 o − 90 o ⇒ ∠ S R Q = 90 o

Now, using the Pythagorean theorem, we will be able to find the value of QR, SR, SP and PQ.
Assume the QC length is "x". Hence the length of RC will also be 'x'. And suppose QR is "y". So, using the Pythagorean theorem, we get: