Selina solutions for Concise Mathematics [English] Class 9 ICSE chapter 12 - Pythagoras Theorem [Proof and Simple Applications with Converse] [Latest edition]

If the lengths of the sides of a triangle are in the ratio 5 : 12 : 13; then the triangle is a ______.

In a right-angled triangle, hypotenuse is 10 cm, and the ratio of the other two sides is 3 : 4; the sides are ______.

ABC is an isosceles triangle with AB = AC = 12 cm and BC = 8 cm. The area of the triangle is ______.

In a rhombus, its diagonals are 30 cm and 40 cm; its perimeter is ______.

In the given figure, AD = 13 cm, DC = 12 cm and BC = 3 cm, then AB is equal to:

A ladder 13 m long rests against a vertical wall. If the foot of the ladder is 5 m from the foot of the wall, find the distance of the other end of the ladder from the ground.

A man goes 40 m due north and then 50 m due west. Find his distance from the starting point.

In the figure: ∠PSQ = 90^o, PQ = 10 cm, QS = 6 cm and RQ = 9 cm. Calculate the length of PR.

In a quadrilateral PQRS, ∠Q = ∠S = 90° then prove that 2PR² – QR² = PQ² + PS² + SR².

AD is drawn perpendicular to base BC of an equilateral triangle ABC. Given BC = 10 cm, find the length of AD, correct to 1 place of decimal.

In triangle ABC, given below, AB = 8 cm, BC = 6 cm and AC = 3 cm. Calculate the length of OC.

In triangle ABC, AB = AC = x, BC = 10 cm and the area of the triangle is 60 cm².
Find x.

If the sides of the triangle are in the ratio 1: `sqrt2`: 1, show that is a right-angled triangle.

Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m;
find the distance between their tips.

In ΔАBC, ∠C = 90° and AC = BC; then AB² is equal to ______.

In the given diagram, AE² + BD² is equal to:

In the given figure, the value of AB × CD is:

ABC is an isosceles triangle right-angled at C. Then 2AC² is equal to ______.

In the figure, given below, AD ⊥ BC.
Prove that: c² = a² + b² - 2ax.

In equilateral Δ ABC, AD ⊥ BC and BC = x cm. Find, in terms of x, the length of AD.

ABC is a triangle, right-angled at B. M is a point on BC.

Prove that: AM² + BC² = AC² + BM²

M andN are the mid-points of the sides QR and PQ respectively of a PQR, right-angled at Q.
Prove that:
(i) PM² + RN² = 5 MN²
(ii) 4 PM² = 4 PQ² + QR²
(iii) 4 RN² = PQ² + 4 QR²(iv) 4 (PM² + RN²) = 5 PR²

In triangle ABC, ∠B = 90^o and D is the mid-point of BC.

Prove that: AC² = AD² + 3CD².

In a rectangle ABCD,
prove that: AC² + BD² = AB² + BC² + CD² + DA².

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

O is any point inside a rectangle ABCD.
Prove that: OB² + OD² = OC² + OA².

In the following figure, OP, OQ, and OR are drawn perpendiculars to the sides BC, CA and AB respectively of triangle ABC.

Prove that: AR² + BP² + CQ² = AQ² + CP² + BR²

Angle AOB is:

Ranbeer runs 10 km due North and 24 km due West. The distance between his two positions is ______.

Angle AOB is ______.

The sides of a rectangle are 12 cm and 16 cm. The length of its diagonal is ______.

Statement (1): ABCD is a rhombus, its diagonal AC = 16 cm and diagonal BD = 12 cm, perimeter of rhombus = 64 cm.

Statement (2): OA = 8 cm, OB = 6 cm. Then, AB = 10 cm and perimeter of rhombus = 40 cm.

Statement (1): Area of given triangle ABC = 6 × 5 cm².

Statement (2): Area of given triangle ABC = `1/2` × 6 × 4 cm².

Assertion (A): Angle BOC = 90°.

Reason (R): OC² = 3² + 4² = 25, OB² = 6² + 8² = 100, OC² + OB² = 125 = BC².

Assertion (A): x = `5sqrt2`.

Reason (R): AC² = 8² + 6² = x² + x²

In the given figure, AB//CD, AB = 7 cm, BD = 25 cm and CD = 17 cm;
find the length of side BC.

In the given figure, ∠B = 90^°, XY || BC, AB = 12 cm, AY = 8cm and AX : XB = 1 : 2 = AY : YC.

Find the lengths of AC and BC.

In ΔABC, ∠B = 90°. Find the sides of the triangle if:

1. AB = (x − 3) cm, BC = (x + 4) cm and AC = (x + 6) cm
2. AB = x cm, BC = (4x + 4) cm and AC = (4x + 5) cm

Each side of rhombus is 10cm. If one of its diagonals is 16cm, find the length of the other diagonals.

In the given figure, diagonals AC and BD intersect at a right angle. Show that:

AB² + CD² = AD² + BC²

Diagonals of rhombus ABCD intersect each other at point O.

Prove that: OA² + OC² = 2AD² - `"BD"^2/2`

In figure AB = BC and AD is perpendicular to CD.
Prove that: AC² = 2BC. DC.

In an isosceles triangle ABC; AB = AC and D is the point on BC produced.

Prove that: AD² = AC² + BD.CD.

In triangle ABC, angle A = 90^o, CA = AB and D is the point on AB produced.
Prove that DC² - BD² = 2AB.AD.

In triangle ABC, AB = AC and BD is perpendicular to AC.

Prove that: BD² − CD² = 2CD × AD

In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1: 3.

Prove that : 2AC² = 2AB² + BC²

In the given figure, AB = 16 cm, BC = 12 cm and CA = 6 cm; find the length of CD.

Gita was feeling hungry, and so she thought to eat something. She looked into the refrigerator and found some bread and cheese. She decided to make cheese sandwiches. She cut the piece of bread diagonally and found that it forms a right-angled triangle with sides containing the right angle are 4 cm and `4sqrt3` cm.

Based on the above information, answer the following:

1. Find the length of the longest side of the sandwich.
2. What is the perimeter of the sandwich?
3. If she wants to wrap the sandwich with silver foil, then what area of foil is needed (assuming 10% extra foil needed for folds)?