B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 11 - Pythagoras Theorem [Latest edition]

In ΔPQR, PS ⊥ QR. Find the sides marked a and b if PR = 41 cm, QS = 12 cm and SR = 40 cm.

The length of a rectangle is 24 cm and its diagonal is 30 cm. Find its breadth and area.

In the following figure, find the sides marked x and y.

In the following figure, find the sides marked x and y.

In the following figure, find the sides marked x and y.

In the following figure, find the sides marked x and y.

In a right-angled triangle, the lengths of sides containing the right angle are 20 cm and 21 cm. What is the radius of the circumcircle of the triangle?

In the given figure, ∠ABC = 90°, AC = 13 cm, CD = 11 cm, AD = 20 cm. Find BC and AB.

In the given figure, PQ and TS are perpendiculars to QS. R is the mid-point of QS and ∠PRT = 90°. If PQ = 9 cm, QS = 24 cm, TS = 16 cm, find PT.

In ΔABD, AB ⊥ BD and AC = BD. If AB = 4 cm, BC = 2 cm, find AD.

In ΔABC, AB = 8, find a and b if a + b = 32.

In the quadrilateral ABCD, ∠B = ∠D = 90°. Prove that 2AC² – BC² = AB² + AD² + DC².

[Hint: LHS = (AC²) + (AC² – BC²) = (AD² + CD²) + (AB²) = RHS]

In triangle ABC, AD ⊥ BC. Prove that (AB + AC)(AC – AB) = (CD + BD)(CD – BD).

The diagonals of quadrilateral PQRS intersect at O, at right angles. Prove that PQ² + SR² = PS² + QR².

In ΔABC, ∠B = 90° and D is the mid-point of BC. Prove that AC² = AD² + 3CD².

In ΔPQR, PS is perpendicular to RQ extended. Prove that PR² = PQ² + QR² + 2QR.SQ.

P is any point inside the rectangle ABCD. Prove that PA² + PC² = PB² + PD².

In ΔABC, AB = AC and CP ⊥ AB. Prove that 

1. 2AB . BP = BC²
2. CP² – BP² = 2AP . BP

In ΔABC, ∠B = 90°. M and N are mid-points of AB and BC respectively.

Prove that

1. CM² + AN² = 5MN²
2. AN² + CM² = AC² + MN²

P is any point inside ΔABC and PQ, PR and PS are drawn perpendiculars to sides AB, BC and AC respectively. Prove that AQ² + BR² + CS² = BQ² + RC² + AS².

In ΔABC, AD is perpendicular to BC and BD : DC = 3 : 1. Prove that `AB^2 = AC^2 + 1/2 BC^2`.

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

[Hint: Draw AP ⊥ BC. Let BD = 2x.
∴ DC = 4x, BC = 6x = AB and DP = x.]

A man travels 21 m in the west direction and then moves 20 m due north. Find his distance from the first point.

A ladder 25 m long reaches a window at a height of 7 m above the ground. Keeping its foot at the same point, the ladder is turned so that it reaches a window 20 m high across the street. Find the total width of the street.

A ladder is placed in such a way that its foot is a distance of 10 m from the wall. If it reaches a window at a height of 24 m, find the length of the ladder.

ABCD is a parallelogram, M is the mid-point of BC and AM ⊥ BC. Prove that AD² = 4(CD² – AM²).

Two poles are 4 m and 9 m high. They are on the same ground and the distance between them is 12 m. Find the distance between their tops.

In ΔABC, ∠B = 90°, AB = BC. BC is produced to a point D. Prove that AD² = 2BC × BD + CD².

PQRS is a rectangle. Prove that PR² + QS² = PQ² + QR² + RS² + PS².

In the rectangle ABCD, AB = 12 cm, BC = 7 cm. P and R are mid-points of AB and CD. AS = BQ = 2.5 cm. Find the perimeter of PQRS.

The upper part of a tree is broken by wind and falls to the ground without being detached. If the top of the broken part touches the ground at a distance of 12 feet from the foot of the tree, calculate the height at which it is broken if the original height is 24 feet.

If the sides of a rectangle are 12 cm, 16 cm, then the length of diagonal is ______.

In ΔABC, AD ⊥ BC, AB = 13 cm, AC = 15 cm, AD = 12 cm.

∴ BC is ______.

In quadrilateral PQRS, ∠Q = 90° = ∠S, SP = 15 cm, PQ = 7 cm, QR = 24 cm.

The length of PR is ______.

In quadrilateral PQRS, ∠Q = 90° = ∠S, SP = 15 cm, PQ = 7 cm, QR = 24 cm.

The length of SR is ______.

In a rhombus the length of the diagonals are 16 cm, 30 cm.

∴ The length of its side is ______.

If the side of a rhombus is 15 cm and one diagonal is 24 cm then the length of the other diagonal is ______.

In ΔABC, AD ⊥ BC. AB = 10 cm, BD = 6 cm, DC = 15 cm.

The value of x is ______.

In ΔABC, AD ⊥ BC. AB = 10 cm, BD = 6 cm, DC = 15 cm.

The value of y is ______.

In the quadrilateral ABCD, AB = 7 cm, BC = 17 cm, AD = CD = x. ∠B = 90° = ∠D.

The value of x is ______.

Which of the sides can not form a right-angled Δ?

A boy cycles 120 m north and moves 160 m west. How far is he from the starting point?

In ΔPQR, PS ⊥ QR, PR = 41 cm, SR = 9 cm, QS = 30 cm.

The value of x is ______.

In ΔPQR, PS ⊥ QR, PR = 41 cm, SR = 9 cm, QS = 30 cm.

The value of y is ______.

The height of pole AB = 8 m. Its shadow on the ground is BC = 6 m.

∴ AC = ______

In ΔABD, AB ⊥ BD and AC = DB. AB = 4 cm, BC = 2 cm.

∴ AD = ______

In the quadrilateral ABCD, ∠B = 90°, ∠ACD = 90°, AB = 3 cm, BC = 4 cm, CD = 12 cm, find AD.

In ΔABC, ∠B = 90°, AD = 17 cm, AB = 15 cm, CD = 12 cm, find AC.

In the figure, find x.

Assertion: In the figure, AC = 17 cm, CD = 7 cm and AE = 15 cm, then ED = 15 cm.

Reason: BC = ED.

Assertion: In the semi circle, O is the centre. OPQR is a rectangle. OP = 6 cm, PQ = 7 cm. The length of PB = 11 cm.

Reason: The radius OQ = `sqrt(85)` cm.

Assertion: In the given figure in ΔABC, ∠B = 90° and DE ⊥ BC. AB = 20 cm, AC = 25 cm = CD. BE = 8 cm, then y = 24 cm.

Reason: BC = 15 cm ∴ x = 7 cm.

Assertion: In the given figure AB = 12 cm, CD = 13 cm and AD = 11 cm. ∴ AC = 20 cm.

Reason: In ΔABC, if ∠B = 90°, then AB² + BC² = AC².

PQRS is a rectangle. PA = 16 cm, AQ = 9 cm and QR = 12 cm.

1. Find the lengths of AS and AR.
2. Prove that ∠SAR = 90°.

ABCD is a rectangle. AP = 6 cm, PB = 5 cm, AQ = 8 cm, QD = 4 cm, DR = 3 cm. Find PQ, QR and PC.

In ΔABC, ∠BAC = 90° = ∠APB. BC = 85 cm, AC = 75 cm and BP = 32 cm. Find x and y.

In ΔABC, AD ⊥ BC. D divides BC in the ratio 2 : 3. Prove that 5AC² = 5AB² + BC².

[Hint: Let BD = 2x, AD = y and DC = 3x.]

ABCD is a rhombus. Prove that AC² + BD² = 4AB².

In the quadrilateral ABCD, ∠B = 90° = ∠D, AB = 7 cm, BC = 17 cm and AD = DC = x. Find the value of x.

AB and CD are two buildings of height 30 m and 63 m. If the distance between them is 56 m, calculate the distance between their tops.

For a standard television set, the horizontal length to the height is in the ratio 4 : 3. Find the length of 35 inches television. [The TV measurement indicates its diagonal length].

In ΔABC, ∠B = 90°, AP = 5, PB = 15, PQ = 17 and QC = 7. Find the values of x and y.

If PQ = 2m, QR = m² – 1 and PR = m² + 1, show that PQR is a right-angled triangle. Hence, find the sides of triangle when m = 4, 5, 6.

ABC is a triangle, right angled at B. M is a point on BC. Prove that AM² + BC² = AC² + BM².