###### Advertisements

###### Advertisements

**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.

###### Advertisements

#### Solution

Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

We consider the rt. angled ΔACD and applying Pythagoras theorem we get,

CD^{2} = AC^{2} + AD^{2}

CD^{2} = AC^{2} + ( AB + BD )^{2} ....[ ∵ AD = AB + BD ]

CD^{2} = AC^{2} + AB^{2} + BD^{2} + 2AB.BD ...(i)

Similarly, in ΔABC,

BC^{2} = AC^{2} + AB^{2}

BC^{2} = 2AB^{2} ...[ AB = AC ]

AB^{2} = `1/2`BC^{2 } ...(ii)

Putting, AB2 from (ii) in (i), We get,

CD^{2} = AC^{2} + `1/2`BC^{2} + BD^{2} + 2AB . BD

CD^{2} - BD^{2} = AB^{2} + AB^{2} + 2AB . ( AD - AB )

CD^{2} - BD^{2} = AB^{2} + AB^{2} + 2AB . AD - 2AB^{2 }

CD^{2} - BD^{2} = 2AB . AD

DC^{2} - BD^{2} = 2AB . AD

Hence Proved.

#### APPEARS IN

#### RELATED QUESTIONS

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`

In a right triangle ABC, right-angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle (in cm) is

(A) 4

(B) 3

(C) 2

(D) 1

Two towers of heights 10 m and 30 m stand on a plane ground. If the distance between their feet is 15 m, find the distance between their tops

In figure, ∠B of ∆ABC is an acute angle and AD ⊥ BC, prove that AC^{2} = AB^{2} + BC^{2} – 2BC × BD

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 AC^{2} = BC × DC

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

The angle B is:

Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

PQR is a triangle right angled at P. If PQ = 10 cm and PR = 24 cm, find QR.

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.

The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.

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

The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is

(A)\[7 + \sqrt{5}\]

(B) 5

(C) 10

(D) 12

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

(3, 5, 4)

**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.**(10, 24, 27)

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 the given figure, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ^{2 }= 4PM^{2 }– 3PR^{2}.

^{}

Walls of two buildings on either side of a street are parellel to each other. A ladder 5.8 m long is placed on the street such that its top just reaches the window of a building at the height of 4 m. On turning the ladder over to the other side of the street , its top touches the window of the other building at a height 4.2 m. Find the width of the street.

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

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

^{}

Digonals of parallelogram WXYZ intersect at point O. If OY =5, find WY.

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

find the length of side BC.

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

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

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

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

Prove that: AM^{2} + BC^{2} = AC^{2} + BM^{2}.

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

Prove that: AR^{2} + BP^{2} + CQ^{2} = AQ^{2} + CP^{2} + BR^{2}

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

Prove that: 2AC^{2} - AB^{2} = BC^{2} + CD^{2} + DA^{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 length of diagonal of the square whose side is 8 cm.

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

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.

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**

6 m, 9 m, and 13 m

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 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 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?

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.

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

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.

9, 40, 41

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?

8, 15, 17

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?

1.5, 1.6, 1.7

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

40, 20, 30

Find the length of the hypotenuse of a triangle whose other two sides are 24cm and 7cm.

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.

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: AB^{2} = AD^{2} - BC x CE + `(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 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: 9AQ^{2 }= 9AC^{2} + 4BC^{2}

In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.

SHow that ∠RTP = 90°.

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.

In a square PQRS of side 5 cm, A, B, C and D are points on sides PQ, QR, RS and SP respectively such as PA = PD = RB = RC = 2 cm. Prove that ABCD is a rectangle. Also, find the area and perimeter of the rectangle.

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

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 __________

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?

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?

Prove that the area of the semicircle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the semicircles drawn on the other two sides of the triangle.

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 ______.

Two trees 7 m and 4 m high stand upright on a ground. If their bases (roots) are 4 m apart, then the distance between their tops is ______.

Two squares having same perimeter are congruent.

Jiya walks 6 km due east and then 8 km due north. How far is she from her starting place?

Two poles of 10 m and 15 m stand upright on a plane ground. If the distance between the tops is 13 m, find the distance between their feet.

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.