What is the perimeter of rectangle ABCD?

Perimeter of rectangle ABCD = 2(AB + BC).

From statement I alone, diagonal BD = 10 units. Since every angle in a rectangle is a right angle, we have a right angled triangle BDA with BD, the hypotenuse, being 10 units in length.

If BD = 10, then AB = 6 and AD = 8. IN this case, the perimeter of rectangle = 2(6+8) = 2(14) = 28.

If BD = 10, AB = 5√2 and AD = 5√2. In this case, the perimeter of rectangle = 2(5√2+ 5√2 ) = 20√2.

Statement I alone is insufficient to find a unique value for the perimeter of the rectangle. Answer options A and D can be eliminated. The possible answer options are B, C or E.

From statement II alone, angle BDC = 30 degrees. This means that angle CBD = 60 degrees since angle BCD = 90 degrees. This tells us that the rectangle is divided into 2 “30-60-90” right angled triangles. This means the sides opposite to these angles i.e. BC, CD and BD are in the ratio of 1: √3 :2. However, knowing the ratio of the sides is not the same as knowing the sides themselves.

Statement II alone is insufficient to find the exact lengths of the sides and hence the perimeter of the rectangle ABCD.
Answer option B can be eliminated. Possible answer options are C or E.

Combining statements I and II, we have the following:

From statement II, we know that the rectangle has been divided into two “30-60-90” right angled triangles. We also know that BC:CD:BD = 1: √3 : 2.
From statement I alone, we know that BD = 10 units.

If BD = 10, BC = 5 units and CD = 5√3 units.

Since ABCD is a rectangle, CD = AB. Therefore, perimeter of rectangle = 2(AB + BC) = 2(5√3 + 5) units.

The combination of statements is sufficient to find the perimeter of the rectangle. Answer option E can be eliminated.
The correct answer option is C.

Hope that helps!