A closed figure having four sides formed by joining four points, no three of which are collinear, in an order is called quadrilateral.
For example:
Quadrilateral is the union of four line segments that join four coplanar points; no three of which are collinear and each segment meet exactly two other lines, each at their end point.
A quadrilateral is convex if each of the internal angle is less than 180˚. Otherwise it is concave quadrilateral.
3.5.2 Properties of quadrilateral
Let ABCD be a quadrilateral.
- Then points A, B, C and D are called the vertices of the quadrilateral.
- The segments AB, BC, CD and DA are the four sides of the quadrilateral.
- The angles ∠DAB, ∠ABC, ∠BCD, and ∠CDA are the four angles of the quadrilateral.
- The segments AC and BD are called as the diagonals of the quadrilateral.
Adjecent sides and opposite sides:
- The sides of a quadrilateral are said to be adjacent sides or consecutive sides. If they have a common end point. In the adjoining figure, AB and AD are adjacent or consecutive sides. Identify the other pair of adjacent sides.
- Two sides of a quadrilateral are said to be opposite angles, if they do not contain a common side. Here ∠DAB and ∠BCD are opposite angles. Identify other pair of opposite angles.
Adjacent angles and opposite angles:
- The two angles of a quadrilateral are adjacent angles or consecutive angles. If they have a common side to them. Thus ∠DAB and ∠ABC are adjacent angles
- Two angles of a quadrilateral are said to be opposite angles, if they do not contain a common side. Here ∠DAB and ∠BCD are opposite angles. Identify other pair of opposite angles.
Diagonal Property:
The diagonal AC divides the quadrilateral into two triangles, namely, triangles ABC and triangles ADC. Name the triangles formed when the diagonal BD is drawn.
Angle Sum Property:
Theorem 1: The sum of the angles of quadrilateral is 360˚
Given: ABCD is a quadrilateral
To prove: ∠A + ∠B + ∠C + ∠D = 360˚
Construction: Draw the diagonal AC
Proof: In triangle ADC, ∠1 + ∠2 + ∠3 = 180˚(angle sum property)
In triangle ABC, ∠4 + ∠5 + ∠6 = 180˚
Adding these two,
∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360˚
But , ∠1 + ∠4 = ∠A and ∠3 + ∠6 = ∠C. Therefore, ∠A + ∠D + ∠B + ∠C = 360˚
Thus, the sum of the angles of quadrilateral is 360˚
Example 1: The four angle of a quadrilateral are in the ratio 2:3:4:6. Find the measures of each angle.
Solution:
Given: The ratio of the angles is 2:3:4:6
To find: The measures of each angle.
Observe that 2 + 3 + 4 + 6 = 15. Thus 15 parts accounts for 360˚. Hence,
15 parts à 360˚
2 parts à (360˚/15) * 2 = 48˚
3 parts à (360˚/15) * 3 = 72˚
4 parts à (360˚/15) * 4 = 96˚
6 parts à (360˚/15) * 6 = 144˚
Thus the angles are 48˚, 72˚, 96˚, 144˚
Example 2: In a quadrilateral ABCD, ∠A and ∠C are of equal measure; ∠B is supplementary to ∠D. Find the measure of ∠A and ∠C.
Solution:
We are given ∠B + ∠D = 180˚. Using angle sum property of a quadrilateral, we get,
∠A + ∠C = 360˚- 180˚ = 180˚
Since, ∠A + ∠C are of equal measure, we obtain ∠A = ∠C = 180˚/2 = 90˚
Example 3: Find all the angles in the given quadrilateral below:
Solution:
We know that, ∠P + ∠Q + ∠R + ∠S = 360˚. Hence,
x + 2x + 3 + x + 3x – 7 = 360˚
This gives, 7x – 4 = 360˚ or 7x = 364˚
Therefore, x = 364˚/7 = 52˚
We obtain ∠P = 52˚; ∠R = 52˚; ∠Q = 2x + 3 = 107˚; ∠S = 149˚. Check that, ∠P + ∠Q + ∠R + ∠S = 360˚
EXERCISE 3.5.2
- Two angles of a quadrilateral are 70° and 130° and the other two angles are equal. Find the measure of these two angles.
Solution:
Let ∠A and ∠B be x, ∠C=70°
and ∠D = 130°
∠A + ∠B + ∠C + ∠D = 360° (Theorem 7)
x + x + 70° + 130° = 360°
2x = 360° − 200°
X = 160°/2 = 80°
∴ ∠A = 80° and ∠B = 80°
- In the figure suppose ∠P and ∠Q are supplementary angles and ∠R = 125°. Find the measures of ∠S.
Solution:
∠P + ∠Q = 180° (Supplementary angles)
∠P + ∠Q + ∠R + ∠S = 360° (Theorem 7)
180° + 125° + ∠S = 360°
∠S = 360°−(180° + 125°)
∠S = 360°−305°
∠S=55°
- Three angles of a quadrilateral are in the ratio 2:3:5 and the fourth angle is
90°. Find the measures of the other three angles.
Solution:
Let the angles be 2x, 3x, 5x.
∠A + ∠B + ∠C + ∠D = 360°
2x + 3x + 5x + 90° = 360°
10x = 360°−90°
x = 270°/10
x = 27°
∴ The angles are , ∠A = 2x = 2×27 = 54°
∠B = 3x = 3×27 = 81
∠C = 5x = 5×27 = 135°
4.In the adjoining figure, ABCD is a quadrilateral such that ∠D + ∠C =100°. The bisectors of ∠A and ∠B meet at ∠P. Determine ∠APB.
Solution:
AP and BP are angular bisectors
∠D + ∠C = 100°
To find: ∠APB
Proof : ∠A + ∠B + ∠C + ∠D = 360° (Theorem 7)
∠A + ∠B + 100° = 360°
∠A + ∠B = 360° − 100°
Multipyling by 12 ,
12 × ∠A + 12 × ∠B = 12 × 260
a + b =130°
In Δ APB,
a + b + ∠P = 180°
∠P = 180° − 130°
∠P = 50°
3.5.3 Trapezium
The set of fundamental have a pair of opposite sides which are parallel. Such a quadrilateral is known as trapezium.
For example:
Example 4: In the figure ABCD, suppose AB||CD; ∠A = 65° and ∠B = 75°. What is the measure of ∠C and ∠D?
Solution:
Observe that, ∠A + ∠D = 180° (a pair of adjacent angles of a trapezium is supplementary.)
Thus, 65° + ∠D = 180°
This gives, ∠D = 180° – 65° = 115°
Similarly, ∠B + ∠C = 180°, which gives 75° + ∠C = 180°. Hence,
∠C = 180° – 75° = 105°
Example 5: In an isosceles trapezium PQRS, ∠P and ∠S are in the ratio 1:2. Find the measure of all angles.
Solution:
In an isosceles trapezium base angles are equal; ∠P = ∠Q. Let ∠P = xº and ∠S = 2xº. Since, ∠P + ∠S = 180º (one pair of adjacent angles of a trapezium is supplementary). We get,
x + 2x = 180º
x = 180º/3 = 60º
Hence, ∠P = 60º and ∠S = 2 x 60º = 120º. Since, ∠P = ∠Q, we get, ∠Q = 60º. But, ∠Q + ∠R = 180º. Hence, we get,
∠R = 180º – 60º = 120º
EXERCISE 3.5.3
1. In a trapezium PQRS, PQ || RS, and ∠P = 70° and ∠Q = 80° . Calculate the measure of ∠S and ∠R.
Solution:
∠P + ∠S = 180° (Supplementary angles) ∠S = 180° − 70° = 110° ∠Q + ∠R = 180° (Supplementary angles)
∠R = 180° −80° = 100°
2. In a trapezium ABCD with AB || CD. It is given that AD is not parallel to BC. Is ΔABC ≅ Δ ADC? Give reasons.
Solution:
In Δ ABC and Δ ADC,
(1) AC = AC (Common side)
(2) ∠BAC = ∠ACD (Alternate angles, AB || CD)
∴ Δ ABC ≅ Δ ADC It cannot be proved with the help of any beam postulated.
3. In the figure, PQRS is an isosceles trapezium; ∠SRP = 30°, and ∠PQS=40° Calculate the angles ∠RPQ and ∠RSQ.
Solution:
Data:PQRS is an isosceles trapezium, PS = RQ and ∠P = ∠Q ∠SRP = 30° and ∠PQS = 40° To find : ∠RPQ and ∠RSQ Proof: ∠RPQ = ∠SRP = 30° (Alternate angles, PQ || SR) ∠RSQ = ∠PRS = 40°
(Alternate angles, PQ || SR)
4. Proof that the base angles of an isosceles trapezium are equal.
Solution:
Data: ABCD is an isosceles trapezium. To Prove: ∠A = ∠B Construction: Join the diagonal BD and AC. Proof : In Δ ACB and ΔBDA (1) BC = AD (Isosceles trapezium) (2)AC = BD (ISosceles trapezium) (3) AB = AB (Common side) ∴ ACB ≅ Δ BDA (SSS postulate) ∴ ∠A = ∠B (Congruency property) 5. Suppose in a quadrilateral ABCD, AC= BD and AD = BC. Prove that ABCD is a trapezium.
Solution:
Data: ABCD is a quadrilateral, AC = BD and AD = BC To prove: ABCD is trapezium Proof: In Δ ADB and Δ BCA (1) AD = BC (Data)
(2) AC = BD (Data)
(3) AB = AB (Common side) ∴ Δ ADB ≅ Δ BCA (SSS postulate) ∴ ∠A = ∠B (Congruency property) AC = BD (Data), AD = BC (Data)
∴ ABCD is an isosceles trapezium
3.5.4 Parallelograms
Look at the following sets of quadrilaterals:
A quadrilateral in which both the pairs of opposite sides are parallel is called a parallelogram.
Proposition 1: In parallelogram, opposite sides are equal and opposite angles are equal.
Proof:
Let ABCD is a parallelogram. Join BD. Then ∠1 = ∠2 and ∠3 = ∠4. In triangles ABD and CBD, we observe that,
∠1 = ∠2 ; ∠4 = ∠3 , BD(common)
Hence, ΔABD ≅ Δ CDB (ASA postulate). It follows that,
AB = DC, AD = BC and ∠A = ∠C.
Similarly join AC, and we can prove that, ΔADC ≅ Δ CBA. Hence, ∠D = ∠B
Example 6: The ratio of two sides of parallelogram is 3:$ and its perimeter is 42cm. Find the measures of all sides of the parallelogram.
Solution:
Let the sides be 3x and 4x. Then the perimeter of the parallelogram is 2(3x + 4x) = 2 x 7x = 4x.
The given data implies that, 42 = 14x, so, that, x = 42/14 = 3. Hence the sides of the parallelogram are 3 x 3 = 9cm and 3 x4 = 12cm
Example 7: In the adjoining figure, PQRS is a parallelogram. Find x and y in cm.
Solution:
In a parallelogram, we know, that the diagonals bisect each other. Therefore SO = OQ. This gives 16 = x + y. Similarly, PO = OR, so that 20 = y + 7. We obtain y = 20 – 7 = 13cm. Substituting the value of y in the first relation, we get 16 = x + 13. Hence x = 3cm
EXERCISE 3.5.4 1.The adjacent angles of a parallelogram are in the ratio 2:1 Find the measures of all the angles.
Solution:
Let the angles be 2x and x ∠A + ∠B = 2x + x = 180° (Adjacent angles of parallelograms are supplementary) 2x + x = 180° 3x = 180° X = 180°/3 = 60° ∴ ∠A = 2x = 60×2 = 120° ∴ ∠B = 60° ∴ ∠C = ∠A = 120° (Opposite angles of parallelogram) ∠D = ∠B = 60°
(Opposite angles of parallelogram)
2. A field is in the form of a parallelogram, whose perimeter is 450 m and one of its sides is larger than the other by 75m. Find the lengths of all sides
Solution:
Perimeter = AD + DC + CB + BA 450 = x + x + 75 + x + x + 75 450 = 4x + 150 450 – 150 = 4x 300𝑥 = x X = 75 ∴ Side = 75m(x)
∴ Opposite side = x + 75 = 75 + 75 = 150m
3. In the figure, ABCD is a parallelogram. The diagonals AC and BD intersect at O; and ∠DAC = 40°, ∠CAB = 35°, and ∠DOC = 10°. ∠ADC, ∠ACB, and ∠CBD.
Solution:
Data: ABCD is a parallelogram AD || BC and DC || AB. The diagonals AC and BD intersect at O ∠DAC = 40°, ∠CAB = 35° and ∠DOC = 110° To find : (1) ∠ABO (2)∠ADC (3)∠ACB (4) ∠CBD Proof: ∠DAC + ∠CAB = ∠A 40° + 35 = ∠A
∠A = 75°
∠C = ∠A = 75° (Opposite angles of parallelogram are equal) ∠D + ∠A = 180° (Supplementary angles) ∠D = 180°−75° = 105° ∠B = ∠D = 105° (Opposite angles of parallelogram are equal) ∠DOC = ∠AOB = 110° (Vertically opposite angles) In Δ AOB, ∠A + ∠O + ∠B = 180° (Sum of all angles of a Δ is 180°) 35° + 110° + ∠B = 180° ∠B = 180°-145° (1) ∠ABO = 35° (2)∠ADC = 105° (Proved) (3)∠ACB = ∠CBD = 40° (Alternate angles, AD || BC) ∠CBD = 105°-35°
(4) ∠CBD = 70°
(1)∠ABO = 35°
(2) ∠ADC = 105°
(3)∠ACB = 40°
(4) ∠CBD = 70° 4. In a parallelogram ABCD, the side DC is produced to E and ∠BCE = 105°.
Calculate and ∠A, ∠B, ∠C and ∠D.
Solution:
∠BCD + ∠BCE = 180° (Linear pair)
∠BCD = 180° − 105° = 75°
In ABCD, ∠A = ∠C = 75° (Opposite angles of parallelogram) ∠ABC = ∠BCE = 105° (Alternate angles, AB || DE) ∠D = ∠B = 105° (Opposite angles of parallelogram)
5. Prove logically the diagonals of a parallelogram bisect each other. Show conversely that a quadrilateral in which diagonals bisect each other is a parallelogram.
Solution:
Data: ABCD is a parallelogram ( ║ln) To Prove : AO = OC and DO = OB Proof: In ΔABO and DOC (1)AB = DC (Opposite sides of angles) (2) ∠AOB = ∠DOC (V.O.A) (3) ∠ABD = ∠BDC (Alternate angles AB ||CD) ∴ ΔABO ≅ ΔDOC (ASA portulate) ∴ AO = OC & DO = BO (Congruence properties) 6. In a parallelogram KLMN, ∟K = 60°. Find the measure of all the angles.
Solution:
∠M = ∠K = 180° (Supplementary angles) ∠N = 180°−60° = 120° ∠M = ∠K = 60° (Opposite angles of parallelogram) ∠L = ∠N = 120° 7. Let ABCD be a quadrilateral in which ∠A = ∠C and ∠B = ∠D. Prove that ABCD is a parallelogram.
Solution:
Solution:
Data: ABCD is a quadrilateral, AB = DC, AD = BC To Prove : ABCD is a parallelogram Construction: Join DB
Proof: In Δ ABD and Δ DBC (1) DB = DB (Common side) (2) AD = BC (Data) (3) AB = DC (Data) ∴ ABD ≅ Δ DBC (SSS postulate) ∴ ∠DBA = ∠BDC (Congruency property) But they are alternate angles These we have AB ║ DC ∴ ∠ADB = ∠DBC (C.P) But they are alternate angles These we have DA ║ BC
∴ ABCD is a parallelogram.
3.5.5 Special kinds of paralellograms
Rectangle:
A rectangle is a parallelogram whose all angles are right angles.
Diagonal properties of a rectangle:
(i) The diagonals of a rectangle are equal
(ii) The diagonals of a rectangle bisect each other.
Example 8: In a rectangle XYWZ, suppose O is the point of intersection of its diagonals. If ∠ZOW = 110º. Calculate the measure of ∠OYW.
Solution:
We know, ∠ZOW = 110º. Hence, ∠WOY = 180º – 110º = 70º (supplementary angles)
Now, OYW is an isosceles triangle, as OY = OW. Hence,
∠OYW = ∠OWY = (180º – 70º)/2 = 110º/2 = 55º
Rhombus:
A rhombus is a parallelogram in which all the four sides are equal.
Example 10: The diagonals of a rhombus are 24cm and 10cm. Calculate the area of the rhombus.
Solution:
We are given that AC = 24cm; BD = 10cm. We know that the diagonals of a rhombus bisect each other at right angles. Let O be the point of intersection of these diagonals. Then we have AO = CO = 12cm and BO = DO = 5cm. We also know that AOD is right angled triangle. Hence the area of triangle AOD is
1/2 x OA x OD = 1/2 x 12 x 5 = 30 cm2
Since a rhombus has four congruent right triangles, is area is 4 x 30 = 120 cm2
Square
A square is a parallelogram, in which
(i) in which all the sides are equal
(ii) each angle is right angle
(iii) diagonals are equal
(iv) diagonals bisect at right angles.
Example 12: A field is in the shape of a sqaure with side 20m. A pathway of 2m width is surrounding it. Find the outer perimeter of the pathway.
Solution:
Width of the pathway is 2m. Length of the side of the outer sqaure = (20 + 2 + 2) = 24m
Hence, perimeter 4 x 24 = 96m
Kite:
Kite is a quadrilateral in which two of isosceles triangles are joined along the common base.
Example 14: In the figure PQRS is a kite; PQ = 3cm and QR = 6cm. Find the perimeter of PQRS.
Solution:
We have PQ = PS = 3cm, QR = SR = 6cm.
Hence, the perimeter = PQ + QR + RS + PS
3 + 6 + 6 + 3 = 18cm
EXERCISE 3.5.5 1. The sides of the rectangle are in the ratio 2:1. The perimeter is 30 cm. Calculate the measure of all the sides.
Solution:
ABCD is a rectangle Let AB:BC = 2:1 ∴ AB = 2x and BC = x Perimeter of the rectangle = 2(1+b) 2(1+b) = 30 2(2x+x) = 30 2×3x = 30 6x = 30 x = 306 = 5 2x = 2×x = 2×5 = 10 1x = 1×4 = 1×5 = 5 AB = 10m; BC = 5cm CD = 10cm; DA = 5cm
2. In the adjacent rectangle ABCD, ∠OCD = 30°. Calculate ∠BOC. What type of triangle is BOC?
Solution:
ABCD is a rectangle
Diagonals AC and BD bisect each other at O and AC = BD.
AO = OC = BO = OD
OCD = ODC = 30°
On ΔCOD, ∠ODC + ∠OCD + ∠COD = 180˚
30° + 30° + ∠COD = 180°
60° + ∠COD = 180°
∴ ∠COD = 180˚ – 60˚ =120°
∠COD + ∠COB = 180˚
∠COB = 180˚ – 120˚
∠COB = 60°
∴ Δ BOC is an isosceles triangle
- All rectangle are parallelograms, but all parallelograms are not rectangles. Justify the statements.
Solution:
All rectangles have all the properties of parallelograms but a parallelogram
May not have all the properties of a rectangle.
- In a rectangle all the angles are right angle but in a parallelogram only opposite angles are equal.
- In a rectangle the diagonals are equal but in a parallelogram diagonals are not equal.
- Prove logically that the diagonals of a rectangle are equal.
Solution:
Data : ABCD is a rectangle.
AC and BD are the diagonals.
To Prove: AC = BD
Proof : In ΔABC and Δ ABD
∠ABC = ∠BAD [90°]
BC = AD [Opp. Sides]
AB = AB [Common side]
∴ ΔABC ≅ ΔABD [SAS]
∴ AC = BD [CPCT]
- The sides of a rectangular park are in the ratio 4:3. If the area is 1728m2 , find the cost of fencing it at the rate of Rs.2.50/m.
Solution:
ABCD is a rectangular park let AB : BC = 4 : 3
∴ AB = 4x and BC = 3x
Area of rectangle = 1×b
4x × 3x = 1728
12x2 = 1728
X2 = 144
X = √144 = 12
Length = 4x = 4×12 = 48m
Breadth = 3x = 3×12 = 36m
Perimeter of a rectangle = 2(l + b)
2(48 + 36) = 2×84 = 168m
Cost of fencing = perimeter ×Rate
= 168 ×2.50
= Rs.420
- A rectangular yard contains two flower beds in the shape of congruent isosceles right triangle. The remaining portion is a yard of trapezoidal.
Shape (see fig) whose parallel sides have lengths 15 m and 25 m. What fraction of the yard is occupied by the flower bed?
Solution:
ABCD is a rectangular yard
AEFB is a trapezium
AB ║ EF AB = 25 m and EF = 15m
AD = BC = 5m
Area of rectangle = ½ × b = 25×5 = 125m² Area of each flower bed = ½ ×b×h = ½ ×5×5 = (25/2) m² Area of both flower beds = 2× (25/2) = 25m²
Fraction of flower beds to yard = 25/125 = 1/5
7. In a rhombus ABCD ∠C = 70°. Find the other angle of the rhombus.
solution:
ABCD is a rhombus
∠C = 70°
But ∠A = ∠C = 70 ∴ ∠A = 70°
∠A + ∠B =180° (adjacent angles) 70 + ∠B = 180° ∠B = 180 − 70 = 110° ∠B = ∠D = 110° ∴ ∠A = 70°; ∠B = 110° ; ∠D = 110°
8. In a rhombus PQRS, ∠SQR = 40° and PQ = 3 cm. Find ∠SPQ, ∠QSR and the perimeter of the rhombus.
Solution:
∠SQR = 40° and PQ = 3cm
∠SQR = 40° and PQ = 3cm ∠PQS = ∠SQR = 40° ∠PQS = 40° (diagonal bisector angles ) 3 cm But ∠PQS = ∠QSP [PQ = PS] ∴ ∠QSP = 40° In Δ PQS, ∠PQS + ∠QSP + ∠SPQ = 180° 40° + 40° + ∠SPQ = 180 ∠SPQ = 180 −80 = 100° ∴ ∠SPQ = 100° In a rhombus PQ = QR = RS = SP = 3cm Perimeter of the Rhombus PQRS = 3×4 = 12cm
9. In a rhombus PQRS, if PQ = 3x -7 and QR = x + 3, find PS.
Solution: In a rhombus all sides are equal PQ = QR = RS = SP ∴ PQ = QR 3x -7 = x + 3 3x -x = 3 + 7 2x = 10 X = 5 PS = x+3 = 5+3 = 8 cm 10. Let ABCD be a rhombus and ∟ABC = 124°. Calculate ∟A, ∟D and ∟C.
In a rhombus, opposite angles are equal
Solution: ∠B = ∠D = 124° ∠A + ∠B =180°
Consecutive angles
∠A + 124° = 180°
∠A + 124° = 180° ∠A = 180-124=56° ∴ ∠A = 56° ; ∠B = 124° and ∠C = 56°
11. Rhombus is a parallelogram: justify.
Solution: Rhombus has all the properties of parallelogram i.e a) Opposite sides are equal and parallel. b) Opposite angles are equal. c) Diagonals bisect each other ∴ Rhombus is a parallelogram. 12. In a given square ABCD, if the area of triangle ABD is 36 cm2 , find (i) the area of triangle BCD;
(ii) the area of the square ABCD.
Solution:
13. The side of a square ABCD is 5 cm and another square PQRS has perimeter equal to 40cm. Find the ratio of the area ABCD to the area of PQRS.
Solution:
Perimeter of the square ABCD = 4 ×side
= 4×5 = 20 Ratio of (perimeter of ABCD)/(Perimeter of PQRS) = 20/40 = 1/2 or 1:2 Area of ABCD = (side)² = (5)² = 25cm Side of PQRS = perimeter/4 = 40/4 = 10cm Area of PQRS = (side)² = (10)² = 100cm²
Ratio of (Area of ABCD)/(Area of PQRS) = 25/100 = 1/4 or 1:4
14. A square field has side 20m. Find the length of the wire required to fence it four times.
Solution: Length of one side of the square = 20 Rs Length of wire required to fence one round = 4×20 Length of wire required to fence four rounds = 80m = 4×80m = 320m
15. List out the differences between square and rhombus.
Solution: Square 1. All the angles are equal 2. Diagonals are equal
3. Area = side × side=(s)²
Rhombus
- Opposite angles are equal
2. Diagonals are unequal
3. Area= 1/2 × Product of diagonals = 1/2 x d1 d2
16. Four congruent rectangles are placed as shown in the figure. Area of the outer square is 4 times that of the inner square. What is the ratio os length to breadth of the congruent rectangles?
Solution:
Let the length of rectangles be ‘a’ and breadth be ‘b’ Side of outer square –(a+b) units Side of outer square (a-b) units Area of outer square = 4 times area of inner square Area of ABCD = 4(Area of PQRS)
(a + b)² = 4(a-b)
(a + b) = 2(a – b)
2a – 2b = (a + b)
2a – a = b + 2b
a = 3b
a/b = 1/3 or 1 : 3