Triangles (1 Mark) (Q.1) in the Figure,

Triangles (1 Mark) (Q.1) In the figure, CD

(A)

(B) -p/b(C)

AB, CD = p. Then

(D) p

(Q.2) ABC is a rt. angled isosceles triangle, rt. angled at B, and AP is bisector of

intersects BC

(1 Mark)

2

at P. Then AC =

(A)

(B)

(C)

(D) 2 BP

(Q.3) In the fig. ABC is a rt. triangle, rt. angled at B. AD and CE are the two medians drawn from A and C respectively. If AC = 5 cm. and AD

(A)

(B)

(C)

(1 Mark)

cm. Then CE

(D) 3 cm

(Q.4) In the given fig. AB || MN, If PA = x – 2, PM = x ; PB = x – 1 and PN = x + 2, find the value of

(1 Mark)

‘x’.

(A) 2 (B) 3(C) 4(D) 5 (Q.5) If three or more parallel lines are intersected by transversals, the intercepts made by them on the transversals are

(1 Mark)

(A) (B) (C) (D) (Q.6) If ABC is an equilateral triangle with side 12 cm, then the area of triangle formed by joined its mid points is : (A)

(B)

(1 Mark)

(C) 64 sq cm(D) 68

(Q.7) If the areas of two similar triangles are 121 cm2 and 64 cm2 respectively and median of the first

(1 Mark)

triangle is 12.1 cm. find the corresponding median of the other.

(A) 8 cm(B) 8.5 cm(C) 8.8 cm(D) 8.9 cm (Q.8) A girl of height 120 cm is walking from the base of a lamp-post at a speed to 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.

(A) 1.6 m (B) 2.2 m(C) 2.4 m(D) 2.6 m (Q.9) The line segment joining the midpoints of any two sides of a triangle is parallel to

(A) right angle (B) isosceles triangle (C) second side (D) Third side (Q.10) Which of the following statement is true? (A) Any two right triangles are similar(B) Any two squares are similar (C) Any two rectangles are similar(D) Two right triangles can not be similar . (Q.11) Given MN || BC, ABC and ANM are _________ .

(1 Mark)

(1 Mark)

(1 Mark) (1 Mark)

(A) similar(B) congruent(C) neither similar nor congruent(D) not similar (Q.12) If D, E, F are the midpoints of sides BC, CA, AB of ABC. Then the DEF and ABC are_____ (A) congruent(B) similar(C) both A & B(D) not similar (Q.13) In ABC, AB > AC and AD BC. Then AB2 – AC2 (A) BD2 + AD2(B) BD2 + CD2(C) BD2 + AC2 (D) BD2 – CD2 (Q.14) If the corresponding sides of two triangles are proportional then they are (A) congruent(B) similar(C) proportional (D) not similar (Q.15) From given fig. express ‘x’ in terms of a, b, c.

(A) (B) (C) (D) ac (b+c ) (Q.16) The areas of two similar triangles are 64 cm2, 49 cm2. Altitude of first one is 6 cm. Then altitude

(1 Mark)

(1 Mark) (1 Mark) Top (1 Mark)

(1 Mark)

of second in cm is.

(A) 5.25cm(B) 3.5(C) 2.56 (D) 2 (Q.17)

ABC and DEF are similar, in which BC = 3.5 cm, EF = 2.5 cm and area of cm. Then area of DEF in sq cm is

(1 Mark) ABC = 7 sq

(A) 4.59 (B) 5.49(C) 9.54 (D) 3.57 (Q.18) If the ratios of areas of two similar triangles are 81 : 49 the ratios of their corresponding angle-bisector segments is: (A) 5 : 4 (B) 9 : 7(C) 4 : 5 (D) 625 : 256 (Q.19) In the figure , ABC is obtuse. Then AC2 =

(1 Mark)

(1 Mark)

(A) AB2 + BC2 – 2BC. BD(B) AB2 + BC2(C) AB2 + BC2 + 2BC. BD(D) AD2 + BD2 (Q.20) The triangle with measurements a = (2p – 1), , c = (2p + 1) is (A) equilateral (B) right angled(C) isosceles(D) scalane triangle (Q.21) In ABC right angled at C, AD is median. Then AB2 = (A) AC2 – AD2(B) AD2 – AC2(C) 3AC2 – 4AD2(D) 4AD2 – 3AC2

(1 Mark) (1 Mark)

(Q.22) In rhombus ABCD, AB2 + BC2 + CD2 + DA2 =

(1 Mark)

(A) OA2 + OB2(B) OB2 + OC2(C) OC2 + OD2(D) AC2 + BD2 (Q.23) The ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding (1 Mark) sides. Then

(A) (Q.24) If

(B) ABC

(C) (D) ab=ad PQR and then find R.

(1 Mark)

(A) 20°(B) 30°(C) 35°(D) 37° (Q.25) A ladder is placed against a wall such that its foot is at a distance of 5.5m from the wall and its

(1 Mark)

top reaches a window 9 m above the ground. Find the length of the ladder.

(A) 10.52 m(B) 10.54 m(C) 11 m (D) 11.9 m (Q.26) In a right angle triangle, one of the angles is 60 degree, the side opposite to this angle is . (A)

(B)

(Q.27) In equilateral triangle ABC, if

(C)

(1 Mark)

(D) (1 Mark)

(A) B) (C) (D) (Q.28) On joining the mid points of the sides of a triangle along with any of the vertices (1 Mark) as the fourth point make a . (A) parallelogram(B) Rhombus(C) rectangle(D) Square. (2 (Q.29) P and Q are points on sides CA and CB respectively of triangle ABC right angled at C. prove that

Marks) [CBSEDELHI 2007]

Top (2 Marks) [CBSEDELHI 2007]

(Q.30) In Fig. 1, DE||AB and FE||DB. Prove that

(3 Marks) [CBSE(Q.31) In Triangle ABC, AD BC and AD2 = BD.DC prove that BAC is a right angle. DELHI 2007] (5 (Q.32) If a line is drawn parallel to one side of a triangle, to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio. Marks) [CBSEUsing the above result, prove the following: DELHI 2007] In Figure DE||BC and BD=CE, prove that ABC is an isosceles triangle.

(Q.33) D is any point on the side BC of a Triangle ABC such that BC.CD.

ADC =

BAC. Prove that CA2=

(2 Marks) [CBSE-

Outside Delhi 2007] (5 (Q.34) Prove that the ratio of the areas of two similar triangles is equal to the ratio of the Marks) square of their corresponding sides. [CBSEUse the above for the following. Outside If the area of two similar triangle are equal prove that they are congruent. Delhi 2007] (5 Marks) (Q.35) If in a triangle, the square on one side is equal to the sum of square on the [CBSEOutside remaining two –sides, prove that the angle opposite to the first side is a right angle. Delhi 2007] (5 (Q.36) In a right angled triangle, prove that the square on the hypotenuse is equal to the sum of the square of the other two sides. Marks) Use the above for the following. [CBSEIn a triangle ABC, AD perpendicular to BC and BD = 3 CD prove that Outside Delhi 2007] (Q.37) In Figure 1,

,

prove that

(3 Marks) [CBSEDELHI 2006]

(6 (Q.38) Prove that the ratio of the areas of two similar triangles is equal to the ratio of the Marks) squares of their corresponding sides. Using the above, prove that the area of the equilateral triangle described on the side of a [CBSEright angled isosceles triangle is half the area of the equilateral triangle described on its DELHI hypotenuse. 2006] (6 (Q.39) Prove that in a right triangle, the square of the hypotenuse is equal to the Marks) sum of the squares of the other two sides. [CBSEUsing the above, in Figure 4, find PR and PQ, when QR = 26cm, P0 = 6cm and OR DELHI = 8 cm. 2006]

(Q.40) In Figure 1,

Prove that AB2 + CD2 = BD2 + AC2.

(3 Marks) [CBSEOutside Delhi 2006]

(6 Marks) (Q.41) Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the [CBSEother two sides. Outside Delhi 2006] (6 Marks) [CBSEQ.42) Making use of the above, prove the following: In Fig. 3, ABCD is a Fig. 3 rhombus. Prove that 4AB2 = AC2 + BD2. Outside Delhi 2006] (3 Marks) [CBSE, the perpendicular BD on hypotenuse AC is drawn. (Q.43) In a right 2 Outside Prove that AC.CD = BC Delhi 2006] (3 (Q.44) In the figure, are on the same base BC. AD and BC Marks) [CBSEDELHI intersect at O. prove that 2005]

(Q.45) Prove that in a right angled triangle the square on the hypotenuse is equal to the sum of the squares on other two sides.

(Q.46) Using the above result, prove that the sum of squares on the sides of a rhombus is equal to sum of squares on its diagonals.

(Q.47) Prove that in a triangle, a line drawn parallel to one side to intersect the other two sides in distinct points, divides the two sides in the same ratio. (Q.48) Using above prove that the quadrilateral ABCD is a trapezium if the diagonals AC and BD of the quadrilateral ABCD intersect each other at O such that

Top (6 Marks) [CBSEDELHI 2005] (6 Marks) [CBSEDELHI 2005] (6 Marks) [CBSEDELHI 2005] (6 Marks) [CBSEDELHI 2005]

(6 Marks) (Q.49) Prove that the ratio of areas of two similar triangles is equal to the ratio of squares of their [CBSEcorresponding sides. DELHI 2006] (6 (Q.50) Apply the above theorem on the following: Marks) 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 [CBSEDELHI cm, QC = 4-5 cm, prove that area of is one-sixteenth of the area of . 2005] (3 Marks) (Q.51) The perpendicular from vertex A on the side BC of triangle ABC intersects [CBSEBC at Point D such that DB = 3 CD. Prove that 2 AB 2 = 2 AC 2 + BC 2 Outside . Delhi 2005]

(Q.52) If a line is drawn parallel to one side of a triangle, prove that the other two sides are divided in the same ratio. Use the above to prove the following: In the given figure DE || AC and DC || AP.

Prove that


.

(6 (Q.53) Prove that the ratio of the areas of two similar triangles is equal to the ratio of the Marks) squares of their corresponding sides. [CBSEUsing the above, do the following: The areas of two similar triangles ABC and DEF are in the ratio of 9 : 16. If BC = 4.5 cm, DELHI 2004] find the length of EF. (6 (Q.54) In a right angled triangle, the square on hypotenuse is equal to the sum of Marks) squares on other two sides. Prove it. [CBSEUse the above to prove the following: DELHI In ABC, AD is perpendicular on BC. Prove that AB2 + CD2 =AC2 + BD2. 2003] (6 (Q.55) Prove that in a right-angled triangle, the square on the hypotenuse is equal to the Marks) sum of the squares on the other two sides. [CBSEUsing the above, prove the following: Outside In ABC, A = 90 , AD BC. Prove that AB2 + CD2 = BD2 + AC2. Delhi 2004] (3 (Q.56) In figure, ABCD is a trapezium in which AB || DC. The diagonals AC and BD Marks) [CBSEintersect at O. Prove that . Outside Delhi 2004]

(Q.57) Prove that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides. Using the above result, prove the following: In a triangle ABC, XY is parallel to BC and it divides triangle into two parts of

equal area. Prove that:


(3 (Q.58) Two triangles ABC and DBC are on the same base BC and on the same side of BC Marks) in which . If CA and BD meet each other at E, show that AE.EC = BE.ED. [CBSE-

(Q.59) If the diagonals of a quadrilateral divide each other proportionally, prove that it is a trapezium. (Q.60) In Fig. 1, D is a point on the side BC of

such that

DELHI 2008] (3 Marks) [CBSEDELHI 2008] Top (2 Marks) [CBSEDELHI 2004]

Prove that AC/AD = CB/AC

(Q.61)

(1 Mark) [CBSEDELHI 2008]

(Q.62) In figure, P and Q are the points on the sides AB and AC respectively of ABC such that AP = 3.5 cm, AQ = 3 cm, PB = 7 cm and QC = 6 cm. If PQ = 4.5 cm, find BC.

(1 Mark) [CBSEDELHI 2008]

(Q.63) Prove that the area of the ratio of two similar triangles is equal to the ratio

(6 Marks) [CBSEOutside Delhi 2008] (3 Marks) [CBSEOutside Delhi

of the squares of their corresponding sides. Using the above, do the following : The diagonals of the trapezium ABCD, with AB|| DC, intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangle AOB to the area of triangle COD.

(Q.64) In figure,

2008]

(3 Marks) (Q.65) D and E are the points on the sides CA and CB respectively of ABC right-angled [CBSEOutside at C. Prove that Delhi 2008] (1 Mark) (Q.66) D, E and F are the mid-points of the sides AB, BC and CA respectively of ABC. [CBSEOutside Find Delhi 2008] (1 Mark) (Q.67) In LMN, L = 500 and N = 600. If LMN PQR, then find Q. [CBSEOutside Delhi 2009] (2 Marks) (Q.68) [CBSEOutside Delhi 2009]

(Q.69)

(Q.70) In figure, M is mid-point of side CD of a parallelogram ABCD. The line BM is drawn intersecting AC at L and AD produced at E. Prove that EL = 2BL

(Q.71) In a triangle, if the square on one side is equal to the sum of the squares on the other two sides, prove that the angle opposite to the first side is a right angle. Use the above theorem to find the measure of PKR in the given figure.

(Q.72) If the areas of two similar triangles are in the ratio 25 : 64, write the ratio of their corresponding sides. (Q.73) In the given figure, M = N = 460. Express x in terms of a, b and c where a, b and c are lengths of LM, MN and NK respectively.


(3 Marks) [CBSEOutside Delhi 2009] (6 Marks) [CBSEOutside Delhi 2009]

(1 Mark) [CBSEOutside Delhi 2009] (1 Mark) [CBSEDELHI 2009]

(Q.74) Q.75) In fig., DEFG is a square and angle BAC = 900. Show that DE2 = BD x EC.

(Q.76)

(Q.77)

(Q.78)

(3 Marks) [CBSEDELHI 2009] Top (3 Marks) [CBSEDELHI 2009]




(Q.79) A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. (Q.80) Prove that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides. (Q.81) Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides Use the above theorem in the figure to prove that

(Q.82) In a trapezium and

in

drawn parallel to AB Cuts

such that

. Diagonal

intersects

at

in

(5 Marks) (5 Marks) (5 Marks)

(4 Marks)

. Prove that

.

(Q.83) A Point O in the interior of a rectangle ABCD is joined with each of the vertices A,B, C and D

(4 Marks)

prove that . Q.84) ABC is a triangle in which AB =AC and D is any point in BC. Prove that

(4 Marks)

.

(Q.85) D, E and F are respectively mid-points of the sides of BC, CA and AB of of the areas of and . (Q.86) In figure is a right triangle, right angled at B, medians and lengths 5 cm and

(Q.87) In figure if

cm. Find the length of

prove that

. Find the ratio are of respective

(4 Marks) (3 Marks)

.

.

(2 Marks)

(Q.88) ABC is a right triangle right-angled at C. Let

and let

be the length

(3 Marks)

of perpendicular form C on AB prove that (i)

(ii)

(Q.89) ABC is a right triangle right-angled at B. Let D and E be any points on AB and BC respectively. (2 Marks) Prove that (Q.90) Any point X inside the

. is joined to its vertices. From a point P in DX, PQ is drawn

parallel to DE meeting XE at Q and QR is drawn parallel to EF meeting XF in R. Prove that (Q.91) The perimeters of two similar triangles are 36 cm and 48 cm respectively. If one side of the first triangles is 9 cm, what is the corresponding side of the other triangle ?

Most Important Questions (Q.1) ABCD is a trapezium in which AB||DC and its diagonal intersect each other at O. Show that (Q.2) In the given figure, E is a point on the side CB produced of an isosceles triangle ABC with AB=AC. If

prove that

Top (2 Marks) (2 Marks)

(Q.3) In the given figure

(Q.4) In the figure ABC and DBC are two triangles on the same base BC. Prove that


(Q.6) In the given figure DE is parallel BC and AD=4x-3, AE = 8x-7, BD=3x-1 and CE= 5x-3, find the value of x.

.

(Q.7) ABC is an equilateral triangle of side 2a. Find each of its altitudes. (Q.8) Give two different examples of pair of (i) Similar Figures (ii) Non-similar figures


(i)

(ii)

(Q.10) Any point X inside

is joined to its vertices. From a point P in DX, PQ is drawn parallel to DE meeting XE at Q and QR is drawn parallel to EF meeting XF in R Prove that

(Q.11) 1 Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of the corresponding medians.

(Q.12) ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Find the ratio of the area of triangles ABC and BDE.

(Q.13) In a trapezium ABCD,AB||DC and DC=2AB; FE drawn parallel to AB cuts AD in F and BC in E, such that

Diagonal DB intersects FE at G.

Prove that 7FE= 10 AB.

(Q.14) In the given figure if XY||AC and XY divides the triangular region ABC into

two parts equal in area. Determine

(Q.15) In the given figure ,M and N are points on sides AB and AC of triangle

ABC such that AM= 4 cm, MB= 8cm AN=6 cm and NC=12 cm. Prove that BC=3MN.

(Q.16) Prove that the equilateral triangles described on the two sides of a rightangled triangle on the hypotenuse in terms of their areas. (Q.17) In figure ABC is a right triangle, right angled at B. Medians AD and CE are of respective lengths 5 cm and

cm. Find the length of Ac

(Q.18) Prove that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides. (Q.19) A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. (Q.20) ABC is a right triangle right-angled at B. Let D and E be any points on AB and BC respectively. Prove that (Q.21) ABC is a right triangle right-angled at C. Let and let be the length of perpendicular form C on AB prove that (i)

(ii)

(Q.22) In Figure if prove that (Q.23) ABC is a triangle in which AB=AC and D is any point in BC. Prove that

Q.24) In the given figure a point O is in the interior of a rectangle ABCD is joined with each of the vertices A,B, C and D prove that

(Q.25) Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides Use the above theorem in the figure to prove that

(Q.26) Two poles of heights 6 m and 11m stand on a plane ground. If the distance between the feet of the poles is 12m, find the distance between their tops. (Q.27) In the given figure O is a point in the interior of the triangle ABC, Show that

Triangles (1 Mark) (Q.1) in the Figure,

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