###### Advertisements

###### Advertisements

In the given figure, point T is in the interior of rectangle PQRS, Prove that, TS^{2 }+ TQ^{2 }= TP^{2 }+ TR^{2 }(As shown in the figure, draw seg AB || side SR and A-T-B)

###### Advertisements

#### Solution

`square`PQRS is a rectangle ...(Given)

∴ PQ || SR ...(1) ...[Opposite sides of a rectangle]

AB || SR ...(2) ...[Construction]

∴ AB || SR || PQ ...[From 1 & 2]

∴ PQ ⊥ PS and PQ ⊥ QR ...(3) ...(Sides of a rectangle)

A line Perpendicular to one of 2 parallel lines is also Perpendicular to the other line

∴ AB ⊥ PS and AB ⊥ QR ...(4)

∴ `square`ABRS is rectangle

∴ AS = BR ...(5) ...[Opposite sides of a rectangle]

Similarly we can prove that AP = BQ ...(6)

Now, AB ⊥ PS and AB ⊥ QR ...[From 4]

∴ ∠SAT = ∠RBT = ∠PAT = ∠QBT = 90° ...[A-T-B]

In ΔPAT, PT^{2} = PA^{2} + AT^{2} ... (7)

In ∆ATS, TS^{2} = AT^{2} + AS^{2 } ...(8)

In ∆QBT, QT^{2} = QB^{2} + BT^{2} ...(9)

In ∆BRT, TR^{2} = BT^{2} + BR^{2} ...(4)

Adding Equations 7 & 8

∴ TS^{2} + TQ^{2}

= AT^{2} + AS^{2} + BT^{2} + BQ^{2}

= AT^{2} + BR^{2} + BT^{2} + AP^{2 }...[From 5 & 6]

= AT^{2} + AP^{2} + BT^{2} + BR^{2}

= TP^{2} + TR^{2} ...[From 9 & 10]

Hence proved.

#### RELATED QUESTIONS

Prove that the diagonals of a rectangle ABCD, with vertices A(2, -1), B(5, -1), C(5, 6) and D(2, 6), are equal and bisect each other.

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

The perpendicular AD on the base BC of a ∆ABC intersects BC at D so that DB = 3 CD. Prove that `2"AB"^2 = 2"AC"^2 + "BC"^2`

Sides of triangles are given below. Determine it is a right triangles? In case of a right triangle, write the length of its hypotenuse. 3 cm, 8 cm, 6 cm

ABC is an isosceles triangle right angled at C. Prove that AB^{2} = 2AC^{2}

In the following figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that

(i) OA^{2} + OB^{2} + OC^{2} − OD^{2} − OE^{2} − OF^{2} = AF^{2} + BD^{2} + CE^{2}

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

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.

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

A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree.

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.

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

**Identify, with reason, if the following is a Pythagorean triplet.**(10, 24, 27)

**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.

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.

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, 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.

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

**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.**

**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 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.

**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}

**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}

**M andN are the mid-points of the sides QR and PQ respectively of a PQR, right-angled at Q.**

Prove that:

(i) PM^{2} + RN^{2} = 5 MN^{2}(ii) 4 PM^{2} = 4 PQ^{2} + QR^{2}(iii) 4 RN^{2} = PQ^{2} + 4 QR^{2}(iv) 4 (PM^{2} + RN^{2}) = 5 PR^{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.

Prove that (1 + cot A - cosec A ) (1 + tan A + sec A) = 2

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**

16 cm, 20 cm, and 12 cm

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: 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?

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

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

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?

11, 60, 61

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

40, 20, 30

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

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 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 }= OA^{2} + OB^{2} + OC^{2} - OD^{2} - OE^{2} - OF^{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} + 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: 9AQ^{2 }= 9AC^{2} + 4BC^{2}

In a right-angled triangle ABC,ABC = 90°, AC = 10 cm, BC = 6 cm and BC produced to D such CD = 9 cm. Find the length of AD.

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} = ________

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 figure, PQR is a right triangle right angled at Q and QS ⊥ PR. If PQ = 6 cm and PS = 4 cm, find QS, RS and QR.

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.

In the adjoining figure, a tangent is drawn to a circle of radius 4 cm and centre C, at the point S. Find the length of the tangent ST, if CT = 10 cm.

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 ______.

Two rectangles are congruent, if they have same ______ and ______.

Two squares are congruent, if they have same ______.

If the areas of two circles are the same, they are congruent.

Two squares having same perimeter 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.