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A quadrilateral has 4 sides, 4 angles and 4 vertices. Each angle and side is of different measures and lengths. But a square is a type of quadrilateral which has all the sides of equal length and every angle should be a right-angle which is 90o. The total sum of interior angles is 360o. To construct a square you need to check that each angle is of 90o.
Read Also: Surface Areas and Formulas
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Key Terms: Square, Angle, Length, Side
Square
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Square is a 2-dimensional figure which has four equal sides and equal angles which should be of 90o which is a right-angle.
The diagonals of a square also have equal lengths and they bisect each other at 90o. The length of the diagonal of a square is more than the sides of a square. If a square is cut from any plane from the center, both the halves we get are always equal.
Square
Check Important Notes for Types of Polygon
Steps for Construction of Square
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How to construct a square when one side is given
Step 1: Draw a line segment AB of the given length and extend the line AB.
Step 2: From point B draw an arc on each side of B of any desired width, name the two points as F and G.
Step 3: From point G draw an arc of any desired width above the point B.
Step 4: Now, without changing the width, draw an arc from point F above the point B, crossing the previous arc, and name the new point H.
Draw a line from B through H
Step 5: Now, take the same width as AB, draw an arc above point A.
Step 6: Without changing the width, from point B draw an arc across BH creating point C - a vertex of the square.
Step 7: Without changing the width, from point C draw an arc to the left of C across the existing arc, creating point D.
Draw the lines CD and AD
ABCD is a square where each side has same length.
Read More: Quadrilateral Angle Sum Property
Steps to Construct Square when Diagonal is Given
Step 1: Construct a line AC as a given diagonal of the square and mark M as a midpoint of the diagonal AC.
Step 2: Construct a perpendicular line at M.
Step 3: Construct a circle which has M as its center and MC as the radius.
Step 4: Name the intersection points between the circle and the perpendicular line at M as B and D.Step 5 - Join the 4 points A, B, C and D to get a square.
Things to Remember
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- Square is 2-dimensional figure which has 4 equal sides. Hence, all the sides are congruent to each other.
- Two diagonals of a square are congruent.
- The length of a diagonal is greater than the side of a square.
- The opposite sides of a square are parallel to each other.
Sample Questions
Ques. Construct a square of side 3cm. (4 Marks)
Ans. According to the question, here are the steps of construction.
Draw a line PQ of 3cm & extend the line PQ.
From point Q draw an arc of any width on each side of Q, creating the two points U and V.
From V draw an arc above the point Q of any width.
Again, draw an arc above Q from point F crossing the previous arc, which would create point X.
Draw a line from Q through X.
Take a width same as of PQ & draw an arc above point P.
Without changing the width, from Q draw an arc across QX creating point R.
Without changing the width, from R draw an arc to the left of R across the exiting arc, creating point S
Draw the lines RSD and PS
PQRS is a square where each side has a length AB.
Ques. Construct a square, each of whose sides measures 6.4 cm. (3 Marks)
Ans. All the sides of a square are equal.
Hence, AB= 6.4cm, ∠A=90o ÂÂ ∠B=90o
Step 1 – Draw AB of 6.4cm and make ∠A & ∠B 90o
Step 2 – Draw and arc of 6.4cm from point A & point B name the points as D & C respectively.
Step 3 - Join C and D.
Ques. Construct a square, each of whose diagonals measures 5.8 cm. (3 Marks)
Ans. The diagonals of a square bisect each other at right angles.
Steps of construction:
Step 1- Draw AC of 5.8 cm
Step 2: Draw a perpendicular bisector XYof AC, with the midpoint as O.
Step 3 – From point O make an arc of 5.8cm on both the sides and name the points B & D.
Step 4: Join AB, BC, AD, DC
Question: Construct a square ABCD whose measurement of the length of the diagonals is 6cm. Measure the sides of a square.
Solution: All the 4 sides of a square are equal and each angle of the square measures 90o.Also diagonals of a square are equal and perpendicularly bisect to each other.
Steps to construct a square whose diagonal is 6cm:
Draw a line AC of 6cm & construct a perpendicular bisector XY with midpoint O.
Therefore, OC = OA= 3cm.
Take O as center & radius 3cm, draw 2 arcs cutting the line XY at points B and D.
Join the segments AB, BC, CD, and DA. Hence, ABCD is the desired square of diagonal 6 cm.
Measuring the sides of a square using a ruler, AB = BC = CD = DA = 4.2 cm.
Ques. A square with side length is inscribed in a circle. Determine the radius of a circle in terms of the side length of square a. (4 Marks)
Ans. The diagonals of a square are the diameter of a circle. Let d and r be the diameter and radius of a circle respectively.
According to the Pythagoras Theorem,
d² = a² + a²
d² = 2a²
d = Ö2a2
d = a√2
The diameter of a circle is twice its radius.
Therefore, r = d/2
r = aÖ2/2
Ques. Choose the correct answer: In a square PQRS, if PQ = (2x + 3) cm and QR = (3x − 5) cm then
(a) x = 4
(b) x = 5
(c) x = 6
(d) x = 8 (3 marks)
Ans. (d) x = 8
All sides of a square are equal.
PQ=QR
(2x+3) = (3x−5)
2x−3x=−5−3
x=8cm
Ques.Let a square have side equal to 6 cm. Find out its area, perimeter and length of diagonal. (2 marks)
Ans. Given; side of the square, s = 6 cm
Area of the square = s2 = 62 = 36 cm2
Perimeter of the square = 4 × s = 4 × 6 cm = 24cm
Length of the diagonal of square = s√2 = 6 × 1.414 = 8.484
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