B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 9 - Inequalities [Latest edition]

In ΔABC, if AB < BC < AC, name the largest and smallest angles of the triangle.

In ΔPQR, ∠P = 80° and ∠Q = 43°. Arrange the sides in descending order of length.

In ΔABC, BC is produced to D so that AB = AC = CD. If ∠BAC = 72°, find the angles of ΔABD and arrange its sides in ascending order of length.

In ΔMPN, MP = 11 cm, PN = 5 cm and MN = 7 cm. Arrange the angles of ΔMPN in ascending order.

Is it possible to draw a triangle whose sides are of the following lengths (all in cm)? Give reason.

6.5, 3.2, 9.7

Is it possible to draw a triangle whose sides are of the following lengths (all in cm)? Give reason.

4.2, 6, 1.7

Is it possible to draw a triangle whose sides are of the following lengths (all in cm)? Give reason.

3.9, 5.2, 3

What are the limits between which the third side of the triangle lies, if the following are its two sides? 

4.8 cm, 6.4 cm

What are the limits between which the third side of the triangle lies if the following are its two sides?

7.2 cm, 5.9 cm

What are the limits between which the third side of the triangle lies, if the following are its two sides? 

6.1 cm, 5.4 cm

Arrange the sides of ΔABC in ascending order of their lengths.

In the given figure, AB is perpendicular to AD and BC. Also ∠C = 40°, ∠D = 30°. Which is longer AO or BO?

In ΔPQR, PQ = PR. QR is extended to a point S. Prove that PS > PQ.

In ΔABC, P is any point inside it. Prove that ∠BPC > ∠A.

[Hint: Extend BP to meet AC in Q. Ext. ∠BPC > Int. opp. ∠PQC and Ext. ∠PQC > ∠A]

Prove that the perimeter of a triangle is greater than the sum of its three medians.

In ΔABC, AB < AC, PB and PC are the bisectors of ∠B and ∠C. Prove that PB < PC.

In a quadrilateral ABCD, P is an internal point. Prove that AP + BP + CP + DP > Semi-perimeter of quadrilateral ABCD.

The sides BA and BC of ΔABC are produced to D and E respectively. Prove that ∠DAC + ∠ECA > 180°.

In ΔPQR, ∠Q = 70°, ∠QPR = 50° and PR = RS. Prove that PQ < PS.

In the quadrilateral ACED, ∠CAD = 100° = ∠D and ∠E = 50°. Show that (i) DE < DB (ii) AB < BC.

AD bisects ∠BAC of ΔABC. Prove that AB > BD.

In the quadrilateral ABCD, prove that AB + BC + CD + AD > AC + BD.

In the quadrilateral PQRS, PQ is the smallest side and SR is the longest side. Prove that ∠P > ∠R and ∠Q > ∠S.

[Hint: Join PR, prove that x > y and a > b. ∴ x + a > y + b, etc.]

In a triangle the lengths of two sides are 4 cm, 8 cm. ∴ The third side can be ______.

In ΔABC, ∠C = 90° then ______.

In ΔPQR, if ∠R > ∠P, then which of the following is true?

In ΔXYZ, ∠X = 100°, ∠Y = 45°.

A triangle can be constructed with sides ______.

In ΔPQR

AD bisects ∠BAC. Sides of ABD arranged in descending order:

In ΔABC, ∠B = 50°, ∠ADB = 80°, AD = DC.

∠C =

In ΔABC, ∠B = 50°, ∠ADB = 80°, AD = DC.

Which of the following is true?

Assertion: In the figure, AC > BC in ΔABC.

Reason: The greater angle has the greater side opposite to it in a triangle.

Assertion: In ΔABC, AB = AC, ∠ACB = 80°, ∠CAD = 25°. AB > BD.

Reason: ∠D = 55°.

Assertion: In a right angled triangle, the hypotenuse is the longest side.

Reason: The other 2 angles of the right triangle are acute.

Assertion: A triangle with sides 3.2 cm, 4.8 cm, 8 cm can be constructed.

Reason: Sum of 2 sides of a triangle is greater than the third side.

In ΔABC, ∠A = 50°, ∠BDC = 110°, ∠ABE = 75° and ∠DBC = x. Find the angle x and arrange the sides of ΔABC in descending order of length.

In the quadrilateral ABCD, AB = AD, BC = CD and BC > AB. Prove that ∠x > ∠y. [Hint: Join AC.]

In ΔPQR, PS ⊥ QR, QT ⊥ PR, ∠P = 74° and ∠R = 36°.

Prove that

1. QO > PO
2. OS > QS

In the quadrilateral ABCD, AC = DC, ∠B = 82°, ∠ACB = 30°. Show that DC > AB and DC > BC.

In a quadrilateral ABCD, AC and BD are diagonals. 

Prove that 

1. AB + BC + CD > AD
2. AB + BC + CD + DA > 2AC
3. AB + BC + CD + DA > 2BD
4. AB + BC + CD + DA > AC + BD

In ΔABD, AC = BC = CD and ∠D = 20°. Find the angles x and y. Hence arrange the sides of ΔABD in ascending order.

In a triangle, show that any side is greater than the difference of the other two sides. [Hint: Construct PS = PR ∴ ∠a = ∠b

In ΔQSR, ext. ∠a > ∠d ⇒ ∠b > ∠d

In ΔPSR, ext. ∠c > ∠b

∠c > ∠a ⇒ ∠c > ∠d

∴ QR > QS

⇒ QR > PQ – PS

QR > PQ – PR]

Side BC of ΔABC is extended to E. DBC is another Δ such that DB intersects AC at P. Find the angles marked x and y and arrange the sides of ΔPCD in ascending order.

AB || DC in quadrilateral ABCD, ∠A = 76°, ∠BDC = 40° and ∠BCE = 125°. Prove that AD < DC.

PQR is an equilateral triangle. S is any point on PQ. Prove that PR > SR.

P is any point inside the triangle ABC. AP is produced to meet BC in Q.

Prove that

1. ∠BPQ > ∠BAQ
2. ∠BPC > ∠BAC