###### Advertisements

###### Advertisements

PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM^{2} = QM . MR

###### Advertisements

#### Solution

Let ∠MPR = x

In ΔMPR

∠MPR = 180º- 90º - x

∠MRP = 90º - x

Similarity, in ΔMPQ

∠MPQ = 90º - ∠MPR

= 90º - x

∠MQP = 180º - 90º - (90º - x)

∠MQP = x

In ΔQMP and ΔPMR

∠MPQ = ∠MRP

∠PMQ = ∠RMP

∠MQP = ∠MPR

∴ΔQMP ~ ΔPMR (By AAA Similarity criterion)

`=>(QM)/(PM) = (MP)/(MR)`

=>PM^{2} = QM x MR

#### APPEARS IN

#### RELATED QUESTIONS

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

ABC is an equilateral triangle of side 2a. Find each of its altitudes.

In an equilateral triangle ABC, D is a point on side BC such that BD = `1/3BC` . Prove that 9 AD^{2} = 7 AB^{2}

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.

ABC is a triangle right angled at C. If AB = 25 cm and AC = 7 cm, find BC.

Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.

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

**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.**

(11, 60, 61)

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 ?

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

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)

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 right angle ΔABC, if ∠B = 90°, AB = 6, BC = 8, then find AC.

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

Find x.

**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 and BD is perpendicular to AC.**

Prove that: BD^{2} - CD^{2} = 2CD × AD.

**In the figure, given below, AD ⊥ BC. **Prove that: c

^{2}= a

^{2}+ b

^{2}- 2ax.

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

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

In the following Figure ∠ACB= 90° and CD ⊥ AB, prove that CD^{2} = BD × AD

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 length of diagonal of the square whose side is 8 cm.

Prove that in a right angle triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

In ∆ ABC, AD ⊥ BC.

Prove that AC^{2} = AB^{2} +BC^{2} − 2BC x BD

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

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

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

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 triplets from among the following set of numbers.

3, 4, 5

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

2, 4, 5

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

2, 6, 7

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?

40, 20, 30

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

A ladder 25m long reaches a window of a building 20m above the ground. Determine the distance of the foot of the ladder from the building.

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

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: 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 : 9(AQ^{2} + BP^{2}) = 13AB^{2}

∆ABC is right-angled at C. If AC = 5 cm and BC = 12 cm. find the length of AB.

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?

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 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

**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?

Foot of a 10 m long ladder leaning against a vertical wall is 6 m away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches.

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).

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

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.

In an equilateral triangle PQR, prove that PS^{2} = 3(QS)^{2}.

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

The longest side of a right angled triangle is called its ______.

Two squares are congruent, if they have same ______.

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?

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.