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? Class 8

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? Class 8

Question 1.

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).

Solution:

(a) ∠CVU = ∠CUV

= [latex]\frac{180^{\circ}-90^{\circ}}{2}[/latex] = 45° .....(i)

Similarly, ∠AUX = ∠AXU = 45°

So, ∠VUX = 180° - 45° - 45° = 90° [From (1)]

Similarly, ∠UXW = ∠XWV

= ∠UVW

= 90° ...(2)

Also, ∆UCV ≅ ∆UAX (By SAS)

So, UV = UX.

Similarly, UX = XW

and XW = WV ...(3)

Thus, from (2) and (3), all the sides and all the angles of UVWX are equal and so it is square.

Note : You can verify it by construction also. Do as directed.

(b) Step 1. With ruler and compass, mark a point D on side PQ with P as centre.

Step 2. Now, with Q as centre and PD as radius, mark a point E on QR (not on PQ).

Step 3. Now, with R as centre and same radius PD, mark a point F on RS (not on QR).

Step 4. Now with S as centre and same radius PD, mark a point G on SP (not on RS).

Step 5. Join DE, EF, FG and GD.

Then, DEFG is the required square.