###### Advertisements

###### Advertisements

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

###### Advertisements

#### Solution

**Given:**AB = 20 cm

∴ AO = `"AB"=20/2` = 10cm

BC = OD = 24 cm

**To find:** Length of AD

In right angled triangle

AOD (AD)^{2} = (AO)^{2} + (OD)^{2}(AD)^{2} = (10)^{2} + (24)^{2}(AD)^{2} = 100 + 576

(AD)^{2} = 676

∴ AD = `sqrt(26xx26)`

AD = 26 cm

#### 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`

The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side is 14 metres more than the shorter side, then find the lengths of the sides of the field.

From a point O in the interior of a ∆ABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove

that :

`(i) AF^2 + BD^2 + CE^2 = OA^2 + OB^2 + OC^2 – OD^2 – OE^2 – OF^2`

`(ii) AF^2 + BD^2 + CE^2 = AE^2 + CD^2 + BF^2`

Sides of triangle are given below. Determine it is a right triangle or not? In case of a right triangle, write the length of its hypotenuse. 7 cm, 24 cm, 25 cm

In Figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that AB^{2} = BC × BD

A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

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

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.

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

**Identify, with reason, if the following is a Pythagorean triplet.**(24, 70, 74)

**Identify, with reason, if the following is a Pythagorean triplet.**

(11, 60, 61)

In the given figure, ∠DFE = 90°, FG ⊥ ED, If GD = 8, FG = 12, find (1) EG (2) FD and (3) EF

Find the length diagonal of a rectangle whose length is 35 cm and breadth is 12 cm.

In ∆PQR, point S is the midpoint of side QR. If PQ = 11, PR = 17, PS = 13, find QR.

Pranali and Prasad started walking to the East and to the North respectively, from the same point and at the same speed. After 2 hours distance between them was \[15\sqrt{2}\]

km. Find their speed per hour.

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 ∆ABC, seg AD ⊥ seg BC, DB = 3CD.

Prove that: 2AB^{2 }= 2AC^{2 }+ 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 right angle ΔABC, if ∠B = 90°, AB = 6, BC = 8, then find AC.

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

**In the given figure, AB//CD, AB = 7 cm, BD = 25 cm and CD = 17 cm;**

find the length of side BC.

**In triangle ABC, AB = AC = x, BC = 10 cm and the area of the triangle is 60 cm ^{2}.**

Find x.

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

Prove that: AD^{2} = AC^{2} + BD.CD.

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

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

**In a quadrilateral ABCD, ∠B = 90° and ∠D = 90°.**

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

**In a rectangle ABCD,**

prove that: AC^{2} + BD^{2} = AB^{2} + BC^{2} + CD^{2} + DA^{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})

Choose the correct alternative:

In right-angled triangle PQR, if hypotenuse PR = 12 and PQ = 6, then what is the measure of ∠P?

Find the side of the square whose diagonal is `16sqrt(2)` cm.

In the given figure, BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL^{2} + CM^{2}) = 5 BC^{2}.

Triangle PQR is right-angled at vertex R. Calculate the length of PR, if: PQ = 34 cm and QR = 33.6 cm.

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

16 cm, 20 cm, and 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 the given figure, angle ACP = ∠BDP = 90°, AC = 12 m, BD = 9 m and PA= PB = 15 m. Find:**(i)** CP**(ii)** PD**(iii)** CD

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

Show that the triangle ABC is a right-angled triangle; if: AB = 9 cm, BC = 40 cm and AC = 41 cm

In the given figure, angle ADB = 90°, AC = AB = 26 cm and BD = DC. If the length of AD = 24 cm; find the length of BC.

A ladder, 6.5 m long, rests against a vertical wall. If the foot of the ladder is 2.5 m from the foot of the wall, find up to how much height does the ladder reach?

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

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

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.

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

4, 5, 6

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.

9, 40, 41

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

8, 15, 17

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

40, 20, 30

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

A right triangle has hypotenuse p cm and one side q cm. If p - q = 1, find the length of third side of the triangle.

Two poles of height 9m and 14m stand on a plane ground. If the distance between their 12m, find the distance between their tops.

In ΔABC, AD is perpendicular to BC. Prove that: AB^{2} + CD^{2} = AC^{2} + BD^{2}

In an equilateral triangle ABC, the side BC is trisected at D. Prove that 9 AD^{2 }= 7 AB^{2}.

From a point O in the interior of aΔABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove that: AF^{2} + BD^{2} + CE^{2} = AE^{2} + CD^{2} + BF^{2}

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: AB^{2} = AD^{2} - BC x CE + `(1)/(4)"BC"^2`

A point OI in the interior of a rectangle ABCD is joined with each of the vertices A, B, C and D. Prove that OB^{2} + OD^{2 }= OC^{2} + OA^{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 : 9(AQ^{2} + BP^{2}) = 13AB^{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.

Determine whether the triangle whose lengths of sides are 3 cm, 4 cm, 5 cm is a right-angled triangle.

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 ________

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

An isosceles triangle has equal sides each 13 cm and a base 24 cm in length. Find its height

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 __________

Rithika buys an LED TV which has a 25 inches screen. If its height is 7 inches, how wide is the screen? Her TV cabinet is 20 inches wide. Will the TV fit into the cabinet? Give reason

In the figure, find AR

From the given figure, in ∆ABQ, if AQ = 8 cm, then AB =?

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 ∆PQR, PD ⊥ QR such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, prove that (a + b)(a – b) = (c + d)(c – d).

The perimeter of the rectangle whose length is 60 cm and a diagonal is 61 cm is ______.

Two circles having same circumference are congruent.

If two legs of a right triangle are equal to two legs of another right triangle, then the right triangles are congruent.

If the hypotenuse of one right triangle is equal to the hypotenuse of another right triangle, then the triangles are congruent.

The hypotenuse (in cm) of a right angled triangle is 6 cm more than twice the length of the shortest side. If the length of third side is 6 cm less than thrice the length of shortest side, then find the dimensions of the triangle.

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?

Points A and B are on the opposite edges of a pond as shown in figure. To find the distance between the two points, the surveyor makes a right-angled triangle as shown. Find the distance AB.

The foot of a ladder is 6 m away from its wall and its top reaches a window 8 m above the ground. Find the length of the ladder.