Nootan solutions for Mathematics [English] Class 9 ICSE chapter 10 - Pythagoras Theorem [Latest edition]

Find whether the given sides of the triangle form a right-angled triangle or not:

3 cm, 4 cm and 5 cm

Find whether the given sides of the triangle form a right-angled triangle or not:

5 cm, 13 cm and 12 cm

Find whether the given sides of the triangle form a right-angled triangle or not:

8 cm, 9 cm and 12 cm

A rectangular garden is 30 m broad and 40 m long. Find the length of its diagonal.

A ladder 13 m long rests against a vertical wall. Its top reaches to a window on the wall at 12 m high. Find the distance of the foot of the ladder from the wall.

Two poles of height 12 m and 24 m stand vertically on a plane ground. If the distance between their tops is 13 m. Find the distance between their feet.

An aeroplane leaves an airport and flies due North at a speed of 200 km\hr. At the same time another aeroplane leaves the same airport and flies due West at a speed of 150 km\hr. How far apart will be the two aeroplanes after 4 hours?.

A ladder 25 m long reaches a window which is 15 m above the ground on one side of the street. When it turned to the other side keeping its foot at the same point, it touches a wall at a height of 20 m from the ground. Find the width of the street.

The side of a rhombus is 10 cm. Its one diagonal is 12 cm. Find the length of other diagonal.

Find the length of the altitude of an equilateral triangle of side 2a cm. 

Find the altitude of an equilateral triangle of side `6sqrt(3)` cm.

In ΔАВC, ∠B = 90°. If AC = (x + 4) cm, BC = (x + 2) cm and AB = (3x + 1) cm, find the sides of triangle.

In the adjoining figure, ∠PQR = 90°, PR = 10 cm, QR = 6 cm and SR = 9 cm. Find PS.

In the adjoining figure, ∠RQS = 90°, ∠QPS = 90°, RS = 25 cm, QR = 20 cm, PQ = 9 cm. Find PS.

In the adjoining figure, DC || AB. BC = 8 cm, AD = 17 cm, CD = 24 cm.

Find

ΔАВС is an isosceles triangle in which AB = AC = 17 cm and BC = 16 cm. Find the length of perpendicular drawn from A to BC.

In the adjoining figure, SR || PQ and ∠PQR = 90°. If PQ = 7 cm, PR = 25 cm and SR = 17 cm, find the length of PS.

The sides of a right-angled triangle are 2x, x + 5 and 3x + 1. If the hypotenuse is 3x + 1, find the sides of triangle.

In ΔАВC, ∠ABC = 90° and D is any point on BC. Prove that : AD² + BC² = AC² + BD².

In ΔPQR, ∠QPR = 90° and PM ⊥ QR. Prove that : PM² = QM.RM.

In a quadrilateral ABCD, ∠B = 90°, AD² = AB² + BC² + CD², prove that ∠ACD = 90°.

ΔАВС is a isosceles triangle in which AB = AC and ∠A = 90°. Prove that : BC² = 2AC².

In ΔABC, ∠A = 90° and BC² = 2AC², prove that : ΔABC is isosceles.

In ΔABC, ∠ABC is an acute angle. Prove that : AC² = AB² + BC² – 2BC.BD.

ΔABC is an equilateral triangle. Side BC is trisected at D. Prove that : 9AD² = 7AB².

In ΔАBC, ∠ABC = 90°. X and Y are mid-points of the sides AB and BC respectively.

Prove that:

1. CX² + AY² = 5XY²
2. 4(CX² + AY²) = 5AC²

In the adjoining figure, AB > AC, BE = EC and ∠ADC = 90°.

Prove that:

1. AB² – AC² = 2BC.ED
2. AB² + AC² = 2(AE² + BE²)

In ΔАВС, AB = AC and D is any point on side BC produced. Prove that: AD² = AB² + BD · CD.

Prove that the sum of the squares on the sides of a rhombus is equal to the sum of square on its diagonals.

The diagonals of a rhombus ABCD intersect each other at O. Prove that: `OA^2 + OC^2 = 2AB^2 - 1/2 BD^2`.

In ◻ABCD, ∠B = 90° and ∠D = 90°. Prove that : 2AC² = AB² + BC² + CD² + DA².

In the adjoining figure, PQ = QR and ∠PSQ = 90°.

Prove that : PR² = 2PQ.RS.

In the adjoining figure, ∠PQR = 90° and XY || QR. If PX : QX = 1 : 2, PQ = 6 cm, PY = 4 cm, find PR and QR.

If O is any point in the interior of a rectangle, ABCD, prove that : OA² + OC² = OB² + OD².

Each side of an equilateral triangle is 10 cm. The length of the each its altitude is ______.

A ladder 17 m long reaches a window above the ground. If the distance of the foot of ladder from wall is 8 m, the height of the window is ______.

The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of the side of rhombus is ______.

In ΔАВC, BC = CA and ∠ACB = 90°. Then correct relation is ______.

In a rhombus ABCD, AC² + BD² is equal to ______.

If the sides of a rectangle are 6 cm and 8 cm then the length of its diagonal is ______.

In ΔABC, ∠ACB > 90°. The correct relation is ______.

In ΔАВС, ∠ACB < 90°. The correct relation is ______.

In ΔABC, AB = AC and BD ⊥ AC then BD² – CD² is equal to ______.

In ΔАВС, ∠ACB = 90°. P and Q are the points on CA and CB respectively which divides these sides in the ratio 2 : 1. Then 9(AQ² + BP²) is equal to ______.