###### Advertisements

###### Advertisements

**Identify, with reason, if the following is a Pythagorean triplet.**(5, 12, 13)

###### Advertisements

#### Solution

In the triplet (5, 12, 13),

5^{2} = 25, 12^{2} = 144, 13^{2} = 169 and 25 + 144 = 169

The square of the largest number is equal to the sum of the squares of the other two numbers.

∴ (5, 12, 13) is a pythagorean triplet.

#### APPEARS IN

#### RELATED QUESTIONS

If the sides of a triangle are 6 cm, 8 cm and 10 cm, respectively, then determine whether the triangle is a right angle triangle or not.

In triangle ABC, ∠C=90°. Let BC= a, CA= b, AB= c and let 'p' be the length of the perpendicular from 'C' on AB, prove that:

1. cp = ab

2. `1/p^2=1/a^2+1/b^2`

ABCD is a rectangle whose three vertices are B (4, 0), C(4, 3) and D(0,3). The length of one of its diagonals is

(A) 5

(B) 4

(C) 3

(D) 25

ABCD is a rhombus. Prove that AB^{2} + BC^{2} + CD^{2} + DA^{2}= AC^{2} + BD^{2}

In a ∆ABC, AD ⊥ BC and AD^{2} = BC × CD. Prove ∆ABC is a right triangle

ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c and let p be the length of perpendicular from C on AB, prove that

(i) cp = ab

`(ii) 1/p^2=1/a^2+1/b^2`

Tick the correct answer and justify: In ΔABC, AB = `6sqrt3` cm, AC = 12 cm and BC = 6 cm.

The angle B is:

In the given figure, ABC is a triangle in which ∠ABC < 90° and AD ⊥ BC. Prove that AC^{2} = AB^{2} + BC^{2} − 2BC.BD.

Which of the following can be the sides of a right triangle?

2.5 cm, 6.5 cm, 6 cm

In the case of right-angled triangles, identify the right angles.

Which of the following can be the sides of a right triangle?

2 cm, 2 cm, 5 cm

In the case of right-angled triangles, identify the right angles.

Which of the following can be the sides of a right triangle?

1.5 cm, 2 cm, 2.5 cm

In the case of right-angled triangles, identify the right angles.

Prove that, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of remaining two sides

For finding AB and BC with the help of information given in the figure, complete following activity.

AB = BC ..........

\[\therefore \angle BAC = \]

\[ \therefore AB = BC =\] \[\times AC\]

\[ =\] \[\times \sqrt{8}\]

\[ =\] \[\times 2\sqrt{2}\]

=

Find the side and perimeter of a square whose diagonal is 10 cm ?

In ∆ABC, AB = 10, AC = 7, BC = 9, then find the length of the median drawn from point C to side AB.

Some question and their alternative answer are given. Select the correct alternative.

If a, b, and c are sides of a triangle and a^{2 }+ b^{2 }= c^{2}, name the type of triangle.

Find the length of the hypotenuse of a right angled triangle if remaining sides are 9 cm and 12 cm.

In ∆ABC, ∠BAC = 90°, seg BL and seg CM are medians of ∆ABC. Then prove that:

4(BL^{2 }+ CM^{2}) = 5 BC^{2}

^{}

In an isosceles triangle, length of the congruent sides is 13 cm and its base is 10 cm. Find the distance between the vertex opposite the base and the centroid.

In a trapezium ABCD, seg AB || seg DC seg BD ⊥ seg AD, seg AC ⊥ seg BC, If AD = 15, BC = 15 and AB = 25. Find A(▢ABCD)

In ΔMNP, ∠MNP = 90˚, seg NQ ⊥ seg MP, MQ = 9, QP = 4, find NQ.

**In ΔABC, Find the sides of the triangle, if:**

- AB = ( x - 3 ) cm, BC = ( x + 4 ) cm and AC = ( x + 6 ) cm
- AB = x cm, BC = ( 4x + 4 ) cm and AC = ( 4x + 5) cm

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

**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.

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

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

Prove that : 2AC^{2} = 2AB^{2} + BC^{2}

**In triangle ABC, angle A = 90 ^{o}, CA = AB and D is the point on AB produced.**

Prove that DC

^{2}- BD

^{2}= 2AB.AD.

**In figure AB = BC and AD is perpendicular to CD.**

Prove that: AC^{2} = 2BC. DC.

**O is any point inside a rectangle ABCD.**

Prove that: OB^{2} + OD^{2} = OC^{2} + OA^{2}.

If P and Q are the points on side CA and CB respectively of ΔABC, right angled at C, prove that (AQ^{2} + BP^{2}) = (AB^{2} + PQ^{2})

If the angles of a triangle are 30°, 60°, and 90°, then shown that the side opposite to 30° is half of the hypotenuse, and the side opposite to 60° is `sqrt(3)/2` times of the hypotenuse.

Find the value of (sin^{2} 33 + sin^{2} 57°)

In Fig. 3, ∠ACB = 90° and CD ⊥ AB, prove that CD^{2} = BD x AD.

Triangle ABC is right-angled at vertex A. Calculate the length of BC, if AB = 18 cm and AC = 24 cm.

Triangle XYZ is right-angled at vertex Z. Calculate the length of YZ, if XY = 13 cm and XZ = 12 cm.

**The sides of a certain triangle is given below. Find, which of them is right-triangle**

6 m, 9 m, and 13 m

In the given figure, angle BAC = 90°, AC = 400 m, and AB = 300 m. Find the length of BC.

In triangle PQR, angle Q = 90°, find: PR, if PQ = 8 cm and QR = 6 cm

In triangle PQR, angle Q = 90°, find: PQ, if PR = 34 cm and QR = 30 cm

In the given figure, angle ACB = 90° = angle ACD. If AB = 10 m, BC = 6 cm and AD = 17 cm, find :

(i) AC

(ii) CD

In the given figure, AD = 13 cm, BC = 12 cm, AB = 3 cm and angle ACD = angle ABC = 90°. Find the length of DC.

A boy first goes 5 m due north and then 12 m due east. Find the distance between the initial and the final position of the boy.

Use the information given in the figure to find the length AD.

In the figure below, find the value of 'x'.

In the right-angled ∆PQR, ∠ P = 90°. If l(PQ) = 24 cm and l(PR) = 10 cm, find the length of seg QR.

In the right-angled ∆LMN, ∠M = 90°. If l(LM) = 12 cm and l(LN) = 20 cm, find the length of seg MN.

The top of a ladder of length 15 m reaches a window 9 m above the ground. What is the distance between the base of the wall and that of the ladder?

Find the Pythagorean triplets from among the following set of numbers.

3, 4, 5

Find the Pythagorean triplet from among the following set of numbers.

2, 6, 7

Find the Pythagorean triplet from among the following set of numbers.

4, 7, 8

The sides of the triangle are given below. Find out which one is the right-angled triangle?

11, 12, 15

The sides of the triangle are given below. Find out which one is the right-angled triangle?

11, 60, 61

From the given figure, find the length of hypotenuse AC and the perimeter of ∆ABC.

A man goes 10 m due east and then 24 m due north. Find the distance from the straight point.

In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC^{2} = AD^{2 }+ BC x DE + `(1)/(4)"BC"^2`

In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC^{2} - AB^{2 }= 2BC x ED

In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB^{2} + AC^{2} = 2(AD^{2} + CD^{2})

In a triangle ABC right angled at C, P and Q are points of sides CA and CB respectively, which divide these sides the ratio 2 : 1.

Prove that: 9BP^{2} = 9BC^{2} + 4AC^{2}

In the given figure. PQ = PS, P =R = 90°. RS = 20 cm and QR = 21 cm. Find the length of PQ correct to two decimal places.

PQR is an isosceles triangle with PQ = PR = 10 cm and QR = 12 cm. Find the length of the perpendicular from P to QR.

A man goes 18 m due east and then 24 m due north. Find the distance of his current position from the starting point?

There are two paths that one can choose to go from Sarah’s house to James's house. One way is to take C street, and the other way requires to take B street and then A street. How much shorter is the direct path along C street?

To get from point A to point B you must avoid walking through a pond. You must walk 34 m south and 41 m east. To the nearest meter, how many meters would be saved if it were possible to make a way through the pond?

The perpendicular PS on the base QR of a ∆PQR intersects QR at S, such that QS = 3 SR. Prove that 2PQ^{2} = 2PR^{2} + QR^{2}

Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels at a speed of `(20 "km")/"hr"` and the second train travels at `(30 "km")/"hr"`. After 2 hours, what is the distance between them?

If in a ΔPQR, PR^{2} = PQ^{2} + QR^{2}, then the right angle of ∆PQR is at the vertex ________

If ‘l‘ and ‘m’ are the legs and ‘n’ is the hypotenuse of a right angled triangle then, l^{2} = ________

In a right angled triangle, the hypotenuse is the greatest side

Find the unknown side in the following triangles

Find the unknown side in the following triangles

Find the distance between the helicopter and the ship

In triangle ABC, line I, is a perpendicular bisector of BC.

If BC = 12 cm, SM = 8 cm, find CS

The hypotenuse of a right angled triangle of sides 12 cm and 16 cm is __________

Find the length of the support cable required to support the tower with the floor

In the figure, find AR

**Choose the correct alternative:**

If length of sides of a triangle are a, b, c and a^{2} + b^{2} = c^{2}, then which type of triangle it is?

In a right angled triangle, if length of hypotenuse is 25 cm and height is 7 cm, then what is the length of its base?

From given figure, In ∆ABC, If AC = 12 cm. then AB =?

Activity: From given figure, In ∆ABC, ∠ABC = 90°, ∠ACB = 30°

∴ ∠BAC = `square`

∴ ∆ABC is 30° – 60° – 90° triangle

∴ In ∆ABC by property of 30° – 60° – 90° triangle.

∴ AB = `1/2` AC and `square` = `sqrt(3)/2` AC

∴ `square` = `1/2 xx 12` and BC = `sqrt(3)/2 xx 12`

∴ `square` = 6 and BC = `6sqrt(3)`

In the given figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:

(i) `"AC"^2 = "AD"^2 + "BC"."DM" + (("BC")/2)^2`

(ii) `"AB"^2 = "AD"^2 - "BC"."DM" + (("BC")/2)^2`

(iii) `"AC"^2 + "AB"^2 = 2"AD"^2 + 1/2"BC"^2`

In an isosceles triangle PQR, the length of equal sides PQ and PR is 13 cm and base QR is 10 cm. Find the length of perpendicular bisector drawn from vertex P to side QR.

Lengths of sides of a triangle are 3 cm, 4 cm and 5 cm. The triangle is ______.

The top of a broken tree touches the ground at a distance of 12 m from its base. If the tree is broken at a height of 5 m from the ground then the actual height of the tree is ______.

In a right-angled triangle ABC, if angle B = 90°, BC = 3 cm and AC = 5 cm, then the length of side AB is ______.

In a right-angled triangle ABC, if angle B = 90°, then which of the following is true?

Jayanti takes shortest route to her home by walking diagonally across a rectangular park. The park measures 60 metres × 80 metres. How much shorter is the route across the park than the route around its edges?

Height of a pole is 8 m. Find the length of rope tied with its top from a point on the ground at a distance of 6 m from its bottom.

The foot of a ladder is 6 m away from its wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its top reach?