A square is drawn inside another square such that each vertex of the inner square lies at the midpoint of a side of the outer square. If the perimeter of the inner square is 40 cm, find the perimeter of the outer square.
\(40\sqrt{2}\) cm
The inner square's perimeter is 40 cm, so its side = \(\frac{40}{4} = 10\) cm.
Each side of the inner square joins the midpoints of two adjacent sides of the outer square, forming the hypotenuse of a right triangle whose legs are each half the outer side \(\frac{a}{2}\).
So inner side \(= \sqrt{\left(\frac{a}{2}\right)^2 + \left(\frac{a}{2}\right)^2} = \frac{a}{\sqrt{2}}\), giving \(\frac{a}{\sqrt{2}} = 10\).
Thus outer side \(a = 10\sqrt{2}\) cm, and outer perimeter \(= 4 \times 10\sqrt{2} = 40\sqrt{2}\) cm.
Hence, the perimeter of the outer square is \(40\sqrt{2}\) cm.
Similar Questions
- If ABCD is a cyclic quadrilateral and ABC is an equilateral triangle find the angle of CDA.
- The angles of a quadrilateral are in the ratio of 1 : 3 : 4 : 7. What is the difference between the largest and the smallest angle?
- If one angle of a parallelogram is $39^\circ$ less than twice the smallest angle, then the smallest angle of the parallelogram is:
- Find the area of a rhombus whose perimeter is 164 cm and one diagonal is of length 80 cm.
- Find the area of a trapezium whose parallel sides are 10 cm and 20 cm and non-parallel sides are equal to 10 cm.
- If one angle of a parallelogram is $18^\circ$ less than twice the smallest angle of the same parallelogram, then the largest angle of the parallelogram is:
- If one angle of a parallelogram is $48^\circ$ less than twice the smallest angle, then the measure of the largest angle of the parallelogram will be:
- If one angle of a parallelogram is $30^\circ$ less than twice the measure of the smallest angle, then the measure of the largest angle of the parallelogram is:
- In a rhombus ABCD, if $\angle ACB = 40^\circ, \text{then } \angle ADB = ?$
- P is the mid-point of side BC of a parallelogram ABCD such that $\angle BAP = \angle DAP$.
If AD = 10 cm, then CD = ?
Important Questions from Quadrilaterals
The ratio between the length and breadth of a rectangular park is 3 : 2. If a man cycling along the boundary at the speed of 12 km per hour completes one round in 8 minutes, then the area of the park in square meter will be
The side of a rhombus is 26 cm. The length of one of its diagonals is 20 cm. The sum of the lengths of the diagonals of this rhombus is equal to the perimeter of a rectangle. If the difference between the length and breadth of the rectangle is 6 cm, then what is the area of the rectangle?
PQRS is a cyclic quadrilateral. If ∠P is 4 times ∠R, and ∠S is 3 times ∠Q, then the average of ∠Q and ∠R is:
ABCD is a trapezium in which AB || DC and DC is perpendicular to BC. If ∠DAB = 110°, then ∠ABC - ∠ADC =_____.
If the quadrilateral has an inscribed circle, then the sum of a pair of opposite sides equals:
