78 solutions available
Fill in the blanks by using the correct word given in brackets. (i) All circles are ……………. . (congruent/similar) (ii) All squares are …………… ....
Give two different examples of pairs of (i) similar figures. (ii) non-similar figures.
State whether the following quadrilaterals are similar or not.
Using B.P.T., prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that...
The diagonals of a quadrilateral ABCD intersect each other at the point O such that \(\frac { AO }{ BO } =\frac { CO }{ D{ O }^{ \bullet } } \) Show...
In the given figure, DE || OQ and DF || OR. Show that EF || QR.
ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that \(\frac { AO }{ BO } =\frac { CO }{ D{ O }^{...
In the given figure, DE || AC and DF || AE. Prove that \(\frac { BF }{ FE } =\frac { BE }{ E{ C }^{ \bullet } } \)
In the given figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.
Using converse of B.P.T., prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that your...
In the given figure (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
E and F are points on the sides PQ and PR respectively of a ∆PQR. For each of the following cases, state whether EF || QR: (i) PE = 3.9 cm, EQ = 3...
In the given figure, if LM || CB and LN || CD. Prove that \(\frac { AM }{ AB } =\frac { AN }{ A{ D }^{ \bullet } } \)
In the given figure, if ∆ABE ≅ ∆ACD, show that ∆ADE ~ ∆ABC.
In the given figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that ∆ABD ~ ∆ECF.
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ∆PQR (see in given figure). Show...
D is a point on the side BC of a triangle ABC, such that ∠ADC = ∠BAC. Show that CA² = CB.CD.
Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that...
A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the...
If AD and PM are medians of triangles ABC and PQR respectively, where ∆ABC ~ ∆PQR. Prove that \(\frac { AB }{ PQ } =\frac { AD }{ P{ M }^{ \bullet }...
State which pairs of triangles in the given figures are similar. Write the similarity criterion used by you for answering the question and also write...
In the given figure, ∆ODC ~ ∆OBA, ∠BOC = 125° and ∠CDO = 70°. Find ∠DOC, ∠DCO and ∠OAB.
Diagonals AC and BD of a trape∠ium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that...
In the given figure, \(\frac { QR }{ QS } =\frac { QT }{ PR } \) and ∠1 = ∠2. show that ∆PQR ~ ∆TQR.
S and T are points on sides PR and QR of ∆PQR such that ∠P = ∠RTS. Show that ∆RPQ ~ ∆RTS.
In the given figure, altitudes AD and CE of ∆ABC intersect each other at the point P. Show that: (i) ∆AEP ~ ∆CDP (ii) ∆ABD ~ ∆CBE (iii) ∆AEP ~ ∆ADB...
E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ∆ABE ~ ∆CFB.
In the given figure, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that:
CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ∆ABC and ∆EFG respectively. If ∆ABC ~ ∆FEG,...
Let ∆ABC ~ ∆DEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.
Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.
In the given figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that: \(\frac { ar\left( ABC \right) }{...
If the areas of two similar triangles are equal, prove that they are congruent.
D, E and F are respectively the mid-points of sides AB, BC and CA of ∆ABC. Find the ratio of the areas of ∆DEF and ∆ABC.
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on...
Tick the correct answer and justify (i) ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles...
ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.
ABC is an equilateral triangle of side la. Find each of its altitudes.
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
In the given figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that (i) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 =...
A ladder 10 m long reaches a window 8 m above the ground. ind the distance of the foot of the ladder from base of the wall.
D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE2 + BD2 = AB2 + DE2.
The perpendicular from A on side BC of a ∆ABC intersects BC at D such that DB = 3CD (see the figure). Prove that 2AB2 = 2AC2 + BC2.
Tick the correct answer and justify : In ∆ABC, AB = 6\(\sqrt { 3 } \)cm, AC = 12 cm and BC = 6 cm. The angle B is: (a) 120° (b) 60° (c) 90° (d) 45
Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse. (i)...
PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM X MR.
In the given figure, ABD is a triangle right angled at A and AC i. BD. Show that (i) AB2 = BC.BD (ii) AC2 = BC.DC (iii) AD2 = BD.CD
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2 , Prove that ABC is a right triangle.
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...
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...
Two poles of heights 6 m and 11m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.
In an equilateral triangle ABC, D is a point on side BC, such that BD = \(\frac { 1 }{ 3 }\)BC. Prove that 9AD2 = 7AB2.
In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.
In the given figure, ABc is triangle in which ∠ABC > 90° and AD ⊥ CB produced. Prove that AC2 = AB2 + BC2 + 2BC X BD
Nazima is fly fishing in a stream. The trip of her fishing rod is 1.8m above the surface of the water and the fly at the end of the string rests on...
In the given figure, PS is the bisector of ∠QPR of ∆PQR. Prove that \(\frac { QS }{ SR } =\frac { PQ }{ PR } \)
In the given figure, D is a point on hypotenuse AC of ∆ABC, DM ⊥ BC and DN ⊥ AB. Prove that: (i) DM2 = DN X MC (ii) DN2 = DM X AN
In the given figure, ABC is atriangle in which ∠ABC 90° and AD ⊥ CB. Prove that AC2 = AB2 + BC2 – 2BC X BD
In the given figure, Ad is a median of a triangle ABC and AM ⊥ BC. Prove that
Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
In the given figure, two chords AB and CD intersect each other at the point P. Prove that: (i) ∆APC ~∆DPB (ii) AP X PB = CP X DP
In the given figure, two chords Ab and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that: (i) ∆PAC ~...
In the given figure, D is a point on side BC of ∆ABC, such that \(\frac { BD }{ CD } =\frac { AB }{ A{ C }^{ \bullet } } \) Prove that AD is the...
