R.S. Aggarwal solutions for मैथमैटिक्स [अंग्रेजी] कक्षा १० chapter 7 - Triangles [Latest edition]

D and E are points on the sides AB and AC respectively of a ΔABC such that DE || BC. If AD = 3.6 cm, AB = 10 cm and AE = 4.5 cm, find EC and AC.

D and E are points on the sides AB and AC respectively of a ΔABC such that DE || BC. If AB = 13.3 cm, AC = 11.9 cm and EC = 5.1 cm, find AD.

D and E are points on the sides AB and AC respectively of a ΔABC such that DE || BC. If `(AD)/(DB) = 4/7` and AC = 6.6 cm, find AE.

D and E are points on the sides AB and AC respectively of a ΔABC such that DE || BC. If `(AD)/(AB) = 8/15` and EC = 3.5 cm, find AE.

D and E are points on the sides AB and AC respectively of a ΔABC such that DE || BC. Find the value of x, when AD = x cm, DB = (x – 2) cm, AE = (x + 2) cm and EC = (x – 1) cm. 

D and E are points on the sides AB and AC respectively of a ΔABC such that DE || BC. Find the value of x, when AD = 4 cm, DB = (x – 4) cm, AE = 8 cm and EC = (3x – 19) cm.  

D and E are points on the sides AB and AC respectively of a ΔABC such that DE || BC. Find the value of x, when AD = (7x – 4) cm, AE = (5x – 2) cm, DB = (3x + 4) cm and EC = 3x cm.

D and E are points on the sides AB and AC respectively of a ΔABC. In the following cases, determine whether DE || BC or not. 

AD = 5.7 cm, DB = 9.5 cm, AE = 4.8 cm and EC = 8 cm. 

D and E are points on the sides AB and AC respectively of a ΔABC. In the following cases, determine whether DE || BC or not. 

AB = 11.7 cm, AC = 11.2 cm, BD = 6.5 cm and AE = 4.2 cm.

D and E are points on the sides AB and AC respectively of a ΔABC. In the following cases, determine whether DE || BC or not. 

AB = 10.8 cm, AD = 6.3 cm, AC = 9.6 cm and EC = 4 cm.

D and E are points on the sides AB and AC respectively of a ΔABC. In the following cases, determine whether DE || BC or not. 

AD = 7.2 cm, AE = 6.4 cm, AB = 12 cm and AC = 10 cm.  

In a ΔABC, AD is the bisector of ∠A. If AB = 6.4 cm, AC = 8 cm and BD = 5.6 cm, find DC. 

In a ΔABC, AD is the bisector of ∠A. If AB = 10 cm, AC = 14 cm and BC = 6 cm, find BD and DC.  

In a ΔABC, AD is the bisector of ∠A. If AB = 5.6 cm, BD = 3.2 cm and BC = 6 cm, find AC. 

In a ΔABC, AD is the bisector of ∠A. If AB = 5.6 cm, AC = 4 cm and DC = 3 cm, find BC.

M is a point on the side BC of a parallelogram ABCD. DM when produced meets AB produced at N. Prove that  

(i) `(DM)/(MN) = (DC)/(BN)` 

(ii) `(DN)/(DM) = (AN)/(DC)` 

Show that the line segment which joins the midpoints of the oblique sides of a trapezium is parallel to the parallel sides.

In the given figure, ABCD is a trapezium in which AB || DC and its diagonals intersect at O. If AO = (5x – 7), OC = (2x + 1), BO = (7x – 5) and OD = (7x + 1), find the value of x.  

In a ΔABC, M and N are points on the sides AB and AC respectively such that BM = CN. If ∠B = ∠C, then show that MN || BC.  

ΔABC and ΔDBC lie on the same side of BC, as shown in the figure. From a point P on BC, PQ || AB and PR || BD are drawn, meeting AC at Q and CD at R respectively. Prove that QR || AD. 

In the given figure, side BC of ΔABC is bisected at D and O is any point on AD. BO and CO produced meet AC and AB at E and F respectively, and AD is produced to X so that D is the midpoint of OX. Prove that AO : AX = AF : AB and show that EF || BC.

ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such that CQ = `1/4` AC. If PQ produced meets BC at R, prove that R is the midpoint of BC.  

In the adjoining figure, ABC is a triangle in which AB = AC. If D and E are points on AB and AC respectively such that AD = AE, show that the points B, C, E and D are concyclic.  

In ΔABC, the bisector of ∠B meets AC at D. A line PQ || AC meets AB, BC and BD at P, Q and R respectively. Show that BP × QR = BQ × PR.

In the given pairs of triangles, find which pair of triangles is similar. State the similarity criterion and write the similarity relation in symbolic form: 

In the given pairs of triangles, find which pair of triangles is similar. State the similarity criterion and write the similarity relation in symbolic form:

In the given pairs of triangles, find which pair of triangles is similar. State the similarity criterion and write the similarity relation in symbolic form:

In the given pairs of triangles, find which pair of triangles is similar. State the similarity criterion and write the similarity relation in symbolic form:

In the given pairs of triangles, find which pair of triangles is similar. State the similarity criterion and write the similarity relation in symbolic form:

In the given figure, ΔODC ~ ΔOBA, ∠BOC = 115° and ∠CDO = 70°. Find (i) ∠DOC (ii) ∠DCO (iii) ∠OAB (iv) ∠OBA.  

In the given figure, ΔOAB ~ ΔOCD. If AB = 8 cm, BO = 6.4 cm, OC = 3.5 cm and CD = 5 cm, find (i) OA (ii) DO.

In the given figure, if ∠ADE = ∠B, show that ΔADE ~ ΔABC. If AD = 3.8 cm, AE = 3.6 cm, BE = 2.1 cm and BC = 4.2 cm, find DE.

The perimeters of two similar triangles ABC and PQR are 32 cm and 24 cm respectively. If PQ = 12 cm, find AB. 

The corresponding sides of two similar triangles ABC and DEF are BC = 9.1 cm and EF = 6.5 cm. If the perimeter of ΔDEF is 25 cm, find the perimeter of ΔABC. 

In the given figure, ∠CAB = 90° and AD ⊥ BC. Show that ΔBDA ~ ΔBAC. If AC = 75 cm, AB = 1 m and BC = 1.25 m, find AD. 

In the given figure, ∠ABC = 90° and BD ⊥ AC. If AB = 5.7 cm, BD = 3.8 cm and CD = 5.4 cm, find BC.

In the given figure, ∠ABC = 90° and BD ⊥ AC. If BD = 8 cm, AD = 4 cm, find CD.

P and Q are points on the sides AB and AC respectively of a ΔABC. If AP = 2 cm, PB = 4 cm, AQ = 3 cm and QC = 6 cm, show that BC = 3PQ.

ABCD is parallelogram and E is a point on BC. If the diagonal BD intersects AE at F, prove that AF × FB = EF × FD.  

In the given figure, DB ⊥ BC, DE ⊥ AB and AC ⊥ BC. Prove that `(BE)/(DE) = (AC)/(BC)`.

A vertical pole of length 7.5 cm casts a shadow 5 m long on the ground and at the same time a tower casts a shadow 24 m long. Find the height of the tower.

In an isosceles ΔABC, the base AB is produced both ways in P and Q such that AP × BQ = AC². Prove that ΔACP ~ ΔBCQ.  

In the given figure, ∠1 = ∠2 and `(AC)/(BD) = (CB)/(CE)`. Prove that ΔACB ~ ΔDCE.

ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the midpoints of AB, AC, CD and BD respectively, show that PQRS is a rhombus.  

In a circle, two chords AB and CD intersect at a point P inside the circle. Prove that (a) ΔPAC ∼ ΔPDB (b) PA · PB = PC · PD.

Two chords AB and CD of a circle intersect at a point P outside the circle. Prove that (a) ΔPAC ~ ΔPDB (b) PA · PB = PC · PD. 

In a right triangle ABC, right angled at B, D is a point on hypotenuse such that BD ⊥ AC. If DP ⊥ AB and DQ ⊥ BC then prove that (a) DQ² = DP · QC (b) DP² = DQ · AP. 

If AD and PM are medians of triangles ABC and PQR, respectively where ΔABC ~ ΔPQR, prove that `(AB)/(PQ) = (AD)/(PM)`.

In the following figure, if ΔABC ∼ ΔDEF and their sides of lengths (in cm) are marked along them, then find the lengths of sides of each triangle.

ΔABC ~ ΔDEF and their areas are respectively 64 cm² and 121 cm². If EF = 15.4 cm, find BC.

The areas of two similar triangles ABC and PQR are in the ratio 9 : 16. If BC = 4.5 cm, find the length of QR.

ΔABC ~ ΔPQR and ar(ΔABC) = 4ar(ΔPQR). If BC = 12 cm, find QR.

The areas of two similar triangles are 169 cm² and 121 cm² respectively. If the longest side of the larger triangle is 26 cm, find the longest side of the smaller triangle.   

ΔABC ~ ΔDEF and their areas are respectively 100 cm² and 49 cm². If the altitude of ΔABC is 5 cm, find the corresponding altitude of ΔDEF.

The corresponding altitudes of two similar triangles are 6 cm and 9 cm respectively. Find the ratio of their areas.

The areas of two similar triangles are 81 cm² and 49 cm² respectively. If the altitude of the first triangle is 6.3 cm, find the corresponding altitude of the other.

The areas of two similar triangles are 64 cm² and 100 cm² respectively. If a median of the smaller triangle is 5.6 cm, find the corresponding median of the other. 

In the given figure, ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 cm, prove that area of ΔAPQ is `1/16` of the area of ΔABC.

In the given figure, DE || BC. If DE = 3 cm, BC = 6 cm and ar(ΔADE) = 15 cm², find the area of ΔABC.

ΔABC is right angled at A and AD ⊥ BC. If BC = 13 cm and AC = 5 cm, find the ratio of the areas of ΔABC and ΔADC.

In the given figure, DE || BC and DE : BC = 3 : 5. Calculate the ratio of the areas of ΔADE and the trapezium BCED. 

In ΔABC, D and E are the midpoints of AB and AC respectively. Find the ratio of the areas of ΔADE and ΔABC. 

The sides of certain triangles are given below. Determine which of them are right triangles.

9 cm, 16 cm, 18 cm 

The sides of certain triangles are given below. Determine which of them are right triangles. 

7 cm, 24 cm, 25 cm

The sides of certain triangles are given below. Determine which of them are right triangles. 

1.4 cm, 4.8 cm, 5 cm

The sides of certain triangles are given below. Determine which of them are right triangles. 

1.6 cm, 3.8 cm, 4 cm

The sides of certain triangles are given below. Determine which of them are right triangles. 

(a – 1) cm, `2sqrta` cm, (a + 1) cm

A man goes 80 m due east and then 150 m due north. How far is he from the starting point?

A man goes 10 m due south and then 24 m due west. How far is he from the starting point? 

A 13 m long ladder reaches a window of a building 12 m above the ground. Determine the distance of the foot of the ladder from the building. 

A ladder is placed in such a way that its foot is at a distance of 15 m from a wall and its top reaches a window 20 m above the ground. Find the length of the ladder. 

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

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, O is a point inside a ΔPQR such that ∠POR = 90°, OP = 6 cm and OR = 8 cm. If PQ = 24 cm and QR = 26 cm, prove that ΔPQR is right-angled.

ΔABC is an isosceles triangle with AB = AC = 13 cm. The length of altitude from A on BC is 5 cm. Find BC.

Find the length of altitude AD of an isosceles ΔABC in which AB = AC = 2a units and BC = a units.

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

Find the height of an equilateral triangle of side 12 cm.

Find the length of a diagonal of a rectangle whose adjacent sides are 30 cm and 16 cm. 

Find the length of each side of a rhombus whose diagonals are 24 cm and 10 cm long. 

In ΔABC, D is the midpoint of BC and AE ⊥ BC. If AC > AB, show that `AB^2 = AD^2 - BC · DE + 1/4 BC^2`.

In the given figure, ∠ACB = 90° and CD ⊥ AB. Prove that `(BC^2)/(AC^2) = (BD)/(AD)`.

In the given figure, D is the midpoint of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that  

(i) `b^2 = p^2 + ax + a^2/4` 

(ii) `c^2 = p^2 - ax + a^2/4`

(iii) `(b^2 + c^2) = 2p^2 + 1/2 a^2` 

(iv) `(b^2 - c^2) = 2ax` 

In ΔABC, AB = AC. Side BC is produced to D. Prove that (AD² − AC²) = BD·CD.

ABC is an isosceles triangle, right-angled at B. Similar triangles ACD and ABE are constructed on sides AC and AB. Find the ratio between the areas of ΔABE and ΔACD. 

An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after `1 1/2`  hours?

In a ΔABC, AD is a median and AL ⊥ BC.

Prove that 

(a) `AC^2 = AD^2 + BC · DL + ((BC)/2)^2` 

(b) `AB^2 = AD^2 - BC · DL + ((BC)/2)^2` 

(c) `AC^2 + AB^2 = 2AD^2 + 1/2 BC^2` 

Naman is doing fly-fishing in a stream. The tip of his fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away from him and 2.4 m from the point directly under the tip of the rod. Assuming that the string (from the tip of his rod to the fly) is taut, how much string does he have out (see the adjoining figure)? If he pulls in the string at the rate of 5 cmcm per second, what will be the horizontal distance of the fly from him after 12 seconds?

State the two properties which are necessary for given two triangles to be similar.

State the basic proportionality theorem.

State and converse of Thale’s theorem. 

State the midpoint theorem.

State the AAA-similarity criterion. 

State the AA-similarity criterion. 

State the SSS-criterion for similarity of triangles.

State the SAS-similarity criterion 

State Pythagoras' theorem.

State the converse of Pythagoras' theorem. 

If D, E, F are the respectively the midpoints of sides BC, CA and AB of ΔABC. Find the ratio of the areas of ΔDEF and ΔABC.

Two triangles ABC and PQR are such that AB = 3 cm, AC = 6 cm, ∠A = 70°, PR = 9 cm, ∠P = 70° and PQ = 4.5 cm. Show that ΔABC ∼ ΔPQR and state the similarity criterion.

In ΔABC ~ ΔDEF such that 2AB = DE and BC = 6 cm, find EF.

In the given figure, DE || BC such that AD = x cm, DB = (3x + 4) cm, AE = (x + 3) cm and EC = (3x + 19) cm. Find the value of x.

A ladder 10 m long reaches the window of a house 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall. 

Find the length of the altitude of an equilateral triangle of side 2a cm. 

ΔABC ~ ΔDEF such that ar(ΔABC) = 64 cm² and ar(ΔDEF) = 169 cm². If BC = 4 cm, find EF.

In a trapezium ABCD, it is given that AB || CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar(ΔAOB) = 84 cm². Find ar(ΔCOD).

The corresponding sides of two similar triangles are in the ratio 2 : 3. If the area of the smaller triangle is 48 cm², find the area of the larger triangle.

In an equilateral triangle with side a, prove that area = `sqrt(3)/4 a^2`.

Find the length of each side of a rhombus whose diagonals are 24 cm and 10 cm long. 

Two triangles DEF and GHK are such that ∠D = 48° and ∠H = 57°. If ΔDEF ∼ ΔGHK then find the measures of ∠F.

In the given figure, MN || BC and AM : MB = 1 : 2. 

Find `("area"(ΔAMN))/("area"(ΔABC))`.

In triangle BMP and CNR it is given that PB = 5 cm, MP = 6 cm, BM = 9 cm and NR = 9 cm. If ΔBMP ∼ ΔCNR then find the perimeter of ΔCNR.

Each of the equal sides of an isosceles triangle is 25 cm. Find the length of its altitude if the base is 14 cm.

A man goes 12 m due south and then 35 m due west. How far is he from the starting point?

If the lengths of the sides BC, CA and AB of a ΔABC are a, b and c respectively and AD is the bisector ∠A then find the lengths of BD and DC.

In the given figure, ∠AMN = ∠MBC = 76°. If p, q and r are the lengths of AM, MB and BC respectively then express the length of MN in terms of p, q and r.  

Find the length of each side of a rhombus are 40 cm and 42 cm. Find the length of each side of the rhombus. 

For the following statement, state whether true (T) or false (F):

Two circles with different radii are similar. 

For the following statement, state whether true (T) or false (F):

Any two rectangles are similar.

For the following statement, state whether true (T) or false (F):

If two triangles are similar then their corresponding angles are equal and their corresponding sides are equal.

For the following statement, state whether true (T) or false (F):

The length of the line segment joining the midpoints of any two sides of a triangle is equal to half the length of the third side.

For the following statement, state whether true (T) or false (F):

In a ΔABC, AB = 6 cm, ∠A = 45° and AC = 8 cm and in a ΔDEF, DF = 9 cm, ∠D = 45° and DE = 12 cm, then ΔABC ∼ ΔDEF.  

For the following statement, state whether true (T) or false (F):

The polygon formed by joining the midpoints of the sides of a quadrilateral is a rhombus. 

For the following statement, state whether true (T) or false (F): 

The ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding medians.

For the following statement, state whether true (T) or false (F): 

If O is any point inside a rectangle ABCD then OA² + OC² = OB² + OD². 

For the following statement, state whether true (T) or false (F):

The sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals.

A man goes 24 m due west and then 10 m due north. How far is he from the starting point?

Two poles of height 13 m and 7 m respectively stand vertically on a plane ground at a distance of 8 m from each other. The distance between their tops is ______.

A vertical stick 1.8 m long casts a shadow 45 cm long on the ground. At the same time, what is the length of the shadow of a pole 6 m high?

A vertical pole 6 m long casts a shadow of length 3.6 m on the ground. What is the height of a tower which casts a shadow of length 18 m at the same time?

The shadow of a 5-m-long stick is 2 m long. At the same time the length of the shadow of a 12.5-m-high tree (in m) is ______.

A ladder 25 m long just reaches the top of a building 24 m high from the ground. What is the distance of the foot of the ladder from the building?

In the given figure, O is a point inside a ΔMNP such that ∠MOP = 90°, OM = 16 cm and OP = 12 cm. If MN = 21 cm and ∠NMP = 90° then NP = ?

The hypotenuse of a right triangle is 25 cm. The other two sides are such that one is 5 cm longer than the other. The lengths of these sides are ______.

The height of an equilateral triangle having each side 12 cm, is ______.

ΔАВС is an isosceles triangle with AB = AC = 13 cm and the length of altitude from A on BC is 5 cm. Then, BC = ?

In a ΔABC it is given that AB = 6 cm, AC = 8 cm and AD is the bisector of ∠A. Then, BD : DC = ?

In a ΔABC it is given that AD is the internal bisector of ∠A. If BD = 4 cm, DC = 5 cm and AB = 6 cm, then AC = ?

In a ΔABC, it is given that AD is the internal bisector of ∠A. If AB = 10 cm, AC = 14 cm and BC = 6 cm, then CD = ?

In a triangle, the perpendicular from the vertex to the base bisects the base. The triangle is ______.

In an equilateral triangle ABC, if AD ⊥ BC then which of the following is true?

In a rhombus of side 10 cm, one of the diagonals is 12 cm long. The length of the second diagonal is ______.

The lengths of the diagonals of a rhombus are 24 cm and 10 cm. The length of each side of the rhombus is ______.

If the diagonals of a quadrilateral divide each other proportionally then it is a ______.

In the given figure, ABCD is a trapezium whose diagonals AC and BD intersect at O such that OA = (3x – 1) cm, OB = (2x + 1) cm, OC = (5x – 3) cm and OD = (6x – 5) cm. Then, x = ?

The line segments joining the midpoints of the adjacent sides of quadrilateral form ______.

If the bisector of an angle of a triangle bisects the opposite side then the triangle is ______.

In ΔABC it is given that `(AB)/(AC) = (BD)/(DC)`. If ∠B = 70° and ∠C = 50° then ∠BAD = ?

In ΔABC, DE || BC so that AD = 2.4 cm, AE = 3.2 cm and EC = 4.8 cm. Then, AB = ?

In a ΔABC, if DE is drawn parallel to BC, cutting AB and AC at D and E respectively such that AB = 7.2 cm, AC = 6.4 cm and AD = 4.5 cm. Then, AE = ?

In ΔABC, DE || BC so that AD = (7x – 4) cm, AE = (5x – 2) cm, DB = (3x + 4) cm and EC = 3x cm. Then, we have

Ιn ΔΑBC, DE || BC such that `(AD)/(DB) = 3/5`. If AC = 5.6 cm then AE = ?

ΔАВС ~ ΔDEF and the perimeters of ΔABC and ΔDEF are 30 cm and 18 cm respectively. If BC = 9 cm then EF = ?

ΔАВС ~ ΔDEF such that AB = 9.1 cm and DE = 6.5 cm. If the perimeter of ΔDEF is 25 cm, what is the perimeter of ΔАВС?

In ΔABC, it is given that AB = 9 cm, BC = 6 cm and CA = 7.5 cm. Also, ΔDEF is given such that EF = 8 cm and ΔDEF ~ ΔABC. Then, perimeter of ΔDEF is ______.

∆ABC and ∆BDE are two equilateral triangles such that D is the mid-point of BC. The ratio of the areas of triangles ABC and BDE is ______.

It is given that ΔАBС ~ ΔDFE. If ∠A = 30°, ∠C = 50°, AB = 5 cm, AC = 8 cm and DF = 7.5 cm then which of the following is true?

In the given figure, ∠BAC = 90° and AD ⊥ BC. Then,

In ΔABC, AB = `6sqrt(3)` cm, AC = 12 cm and BC = 6 cm. Then, ∠B is ______.

In ΔABC and ΔDEF, it is given that `(AB)/(DE) = (BC)/(FD)` then ______.

In ΔDEF and ΔPQR, it is given that ∠D = ∠Q and ∠R = ∠E, then which of the following is not true?

If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true?

In ΔABC and ΔDEF, it is given that ∠B = ∠E, ∠F = ∠C and AB = 3DE, then the two triangles are ______.

If in two triangles ABC and PQR, `(AB)/(QR) = (BC)/(PR) = (CA)/(PQ)`, then ______.

In the given figure, two line segments AC and BD intersect each other at the point P such that PA = 6 cm, PB = 3 cm, PC = 2.5 cm, PD = 5 cm, ∠APB = 50° and ∠CDP = 30° then ∠PBA = ?

Corresponding sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio

It is given that ΔАВС ~ ΔРQR and `(BC)/(QR) = 2/3` then `(ar(ΔPQR))/(ar(ΔABC))` = ?

In an equilateral ∆ABC, D is the midpoint of AB and E is the midpoint of AC. Then, ar(∆ABC): ar(∆ADE) = ?

In ΔABC and ΔDEF, we have `(AB)/(DE) = (BC)/(EF) = (AC)/(DF) = 5/7`, then ar(ΔABC) : ar(ΔDEF) = ?

ΔАВС ~ ΔDEF such that ar(ΔАBC) = 36 cm² and ar(ΔDEF) = 49 cm². Then, the ratio of their corresponding sides is ______.

Two isosceles triangles have their corresponding angles equal and their areas are in the ratio 25 : 36. The ratio of their corresponding heights is ______.

The line segments joining the midpoints of the sides of a triangle form four triangles, each of which is ______.

If ΔABC ~ ΔQRP, `(ar(ΔABC))/(ar(ΔPQR)) = 9/4`, AB = 18 cm and BC = 15 cm then PR = ?

In the given figure, O is the point of intersection of two chords AB and CD such that OB = OD and ∠AOC = 45°. Then, ΔОAC and ΔODB are ______.

In an isosceles ΔАBC, if AC = BC and AB² = 2AC² then C = ?

In ΔABC, if AB = 16 cm, BC = 12 cm and AC = 20 cm, then ΔABC is ______.

Which of the following is a true statement?

Which of the following is a false statement?

Match the following columns:

Column I	Column II
(a) In a given ΔABC, DE || BC and `(AD)/(DB) = 3/5`. If AC = 5.6 cm then AE = ..... cm.	(p) 6
(b) If ΔABC ~ ΔDEF such that 2AB = 3DE and BC = 6 cm then EF = ...... cm.	(q) 4
(c) If ΔABC ~ ΔPQR such that  ar(ΔABC) : ar(ΔPQR) = 9 : 16 and ВС = 4.5 cm then QR = ...... cm.	(r) 3
(d) In the given figure, AB || CD and OA = (2x + 4) cm, OB = (9x – 21) cm,  OC = (2x – 1) cm and OD = 3 cm. Then x = ?	(s) 2.1

Match the following columns:

ΔАВС ~ ΔDEF and their perimeters are 32 cm and 24 cm respectively. If AB = 10 cm then DE = ?

In the given figure, DE || BC. If DE = 5 cm, BC = 8 cm and AD = 3.5 cm then AB = ?

Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their feet is 12 m then the distance between their tops is ______.

The areas of two similar triangles are 25 cm² and 36 cm² respectively. If the altitude of the first triangle is 3.5 cm then the corresponding altitude of the other triangle is ______.

If ΔАВC ~ ΔDEF such that 2AB = DE and BC = 6 cm, find EF.

In the given figure, DE || BC such that AD = x cm, DB = (3x + 4) cm, AE = (x + 3) cm and EC = (3x + 19) cm. Find the value of x.

A ladder 10 m long reaches the window of a house 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall. 

Find the length of the altitude of an equilateral triangle of side 2a cm. 

ΔABC ~ ΔDEF such that ar(ΔABC) = 64 cm² and ar(ΔDEF) = 169 cm². If BC = 4 cm, find EF.

In a trapezium ABCD, it is given that AB || CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar(ΔAOB) = 84 cm². Find ar(ΔCOD).

The corresponding sides of two similar triangles are in the ratio 2 : 3. If the area of the smaller triangle is 48 cm², find the area of the larger triangle.

In the given figure, LM || CB and LN || CD. Prove that `(AM)/(AB) = (AN)/(AD)`.

Prove that the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

In an equilateral triangle with side a, prove that area = `sqrt(3)/4 a^2`.

Find the length of each side of a rhombus whose diagonals are 24 cm and 10 cm long. 

Prove that the ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding sides.

In the given figure, ΔABC and ΔDBC have the same base BC. If AD and BC intersect at O, prove that `(ar(ΔABC))/(ar(ΔDBC)) = (AO)/(DO)`.

In the given figure, XY || AC and XY divides ΔABC into two regions, equal in area. Show that `(AX)/(AB) = ((2 - sqrt(2)))/2`.

In the given figure, ∆ABC is an obtuse triangle, obtuse-angled at B. If AD ⊥ CB (produced) prove that AC² = AB² + BC² + 2BC · BD.

In the given figure, each one of PA, QB and RC is perpendicular to AC. If AP = x, QB = z, RC = y, AB = a and BC = b, show that `1/x + 1/y = 1/z`.