CASE is a square. The points U, V, W, and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square, as shown in Figure (b).

Question

Bright Tutorials · Accepted Answer

(a) U, V, W, and X are the midpoints of the sides of the square.
In ∆VCU and ∆UAX,
we have VC = UA, ∠VCU = ∠UAX = 90°, and CU = AX.
∴ By the SAS condition, ∆VCU and ∆UAX are congruent.
∴ VU = UX
Similarly, VU = XW, VU = WV.
∴ Sides of the quadrilateral UVWX are equal.
In ∆VCU, VC = CU
⇒∠1 = ∠2
Also, ∠1 + ∠C + ∠2 = 180°
⇒∠1 + 90° + ∠1 = 180°
⇒2∠1 = 90°
⇒∠1 = 45°
∴ ∠2 is also 45°.
Similarly, ∠3 = ∠4 = 45°
We have ∠2 + ∠VUX + ∠3 = 180°
⇒45° + ∠VUX + 45° = 180°
⇒∠VUX = 180° – 90°
⇒ ∠VUX = 90°
Similarly, ∠VXW = 90°, ∠XWV = 90° and ∠WVU = 90°.
∴ By definition, the quadrilateral UVWX is a square.
(b) Let ABCD be a square.
Take points P, Q, R, and S such that AS = BP = CQ = DR.
Since the sides of squares are equal,
we have DS = AP = BQ = CR.
In ∆PAS and ∆SDR, we have
PA = SD, ∠PAS = ∠SDR = 90°, and AS = DR.
∴ By the SAS condition, ∆PAS and ∆SDR are congruent.
∴ PS = SR
Similarly, PS = RQ, PS = QP.
∴ Sides of the quadrilateral PQRS are equal.
In ∆PAS, ∠1 + ∠2 + 90° = 180°
⇒∠1 + ∠2 = 90°
⇒∠3 + ∠2 = 90° (∵ ∠1 = ∠3)
Also, ∠2 + ∠4 + ∠3 = 180°
⇒ 90° + ∠4 = 180°
⇒ ∠4 = 180° – 90°
⇒ ∠4 = 90°
∴ Similarly, ∠5 = 90°, ∠6 = 90°, and ∠7 = 90°.
By definition, the quadrilateral PQRS is a square.

Quadrilaterals — Question 7