Criteria for Similarity of Triangles

Q.

A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 $\frac{m}{s}$. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. [4 MARKS]

Q.

In fig., PQ is parallel to MN. If $\frac{KP}{PM}=\frac{4}{13}$ and KN = 20.4 cm. Find KQ [2 MARKS]

Q.

Which of the following are correct?

1. All equilateral triangles are always similar.

2. Similar triangles are always congruent.

3. Congruent triangles are always similar.

4. All circles are similar.

Q.

In $\Delta ABC$, the bisector of $\angle B$ meets AC at D. A line PQ || AC meets AB, BC and BD at P, Q and R respectively.

Show that PR $\times$ BQ = QR $\times$ BP.

Q. (a) A vertical stick 12 cm long casts a shadow 8 cm long on the ground. At the same time, a tower casts a shadow 40 m long on the ground. Determine the height of the tower.

(b) The perimeter of two similar triangles are 30 cm and 20 cm, respectively, If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle. [6 MARKS]

Q.

In the given figure, if $\angle ADE=\angle B$, show that $\Delta ADE\sim \Delta ABC$. If AD = 3.8 cm, AE = 3.6 cm, BE = 2.1 cm and BC = 4.2 cm, find DE.

Q. If $\Delta PQR\sim \Delta MNO$, then which of the following is always true?

1. $\angle PQR=\angle MON$
2. $\angle PRQ=\angle MNO$
3. $\angle RQP=\angle ONM$
4. $\angle QRP=\angle MNO$

Q.

The ratio of the corresponding sides of two similar triangles is 1:3. The ratio of their corresponding heights is –

1. 3:1

2. 1:3

3. 1:9

4. 9:1

Q.

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.

D and E are points on the sides AB and AC respectively of a $\Delta ABC$. In each of the follwing cases, determine whether DE || BC or not.

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

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

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

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

Q.

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

(i) $\frac{DM}{MN}=\frac{DC}{BN}$

(ii) $\frac{DN}{DM}=\frac{AN}{DC}$.

Q.

D and E are points on the sides AB and AC respectively of a $△$ABC such that DE || BC

and divides $△$ABC into two parts, equal in area. Find $\frac{BD}{AB}$

Q.

D and E are respectively the points on the sides AB and AC of a $\Delta ABC$ such that AB = 5.6 cm. AD = 1.4 cm. AC = 7.2 cm and AE = 1.8 cm. Show that $DE\parallel BC$. [1 MARK]

Q. In the given figure, AB and DE are perpendicular to BC. If AB = 9 cm, DE = 3 cm and AC = 24 cm, calculate AD. [4 MARKS]

Q.

Question 2 (iii)
E and F are points on the sides PQ and PR respectively of a $\Delta PQR$. For the following case, state whether EF || QR.
(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.63 cm

Q.

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

Q. If in ∆ABC and ∆DEF, $\frac{\mathrm{AB}}{\mathrm{DE}}=\frac{\mathrm{BC}}{\mathrm{FD}}$, then ∆ABC ∼ ∆DEF when

(a) ∠A = ∠F
(b) ∠A = ∠D
(c) ∠B = ∠D
(d) ∠B = ∠E

Q.

In the given figure, $\Delta OAB\sim \Delta OCD$. If AB = 8 cm, BO = 6.4 cm, OC = 3.5 cm and CD = 5 cm, find (i) OA (ii) DO.

Q.

Which all statements are true :

(i) Two circles with different radii are similar.

(ii) Any two rectangles are similar.

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

(iv) 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.

(v) In a $\Delta ABC,AB=6cm,\angle A={45}^{o}andAC=8cm$ and in a $\Delta DEF$ DF = 9 cm, $\angle D={45}^o$ and DE = 12 cm, then $\Delta ABC\sim \Delta DEF$.

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

(vii) The ratio of the areas of two similar triangles is equal to the ratio of their corresponding angle-bisector segments.

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

(ix) If O is any point inside a rectangle ABCD then $O{A}^2+O{C}^2=O{B}^2+O{D}^2$.

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

Q. Assertion (A): If one angle of a triangle is equal to one angle of another triangle and bisectors of these angles divide the opposite sides in the same ratio, then the triangles are similar.
Reason (R): The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

Which of the following is true?

1. (A) is true and (R) is false
2. (A) and (R) are true but (R) is not the correct explanation of (A)
3. Both (A) and (R) are false
4. (A) and (R) are true and (R) is the correct explanation of (A)

Q.

The largest angle of a triangle, whose sides are 12, 18 and 20 inches, is bisected. Find the lengths of the segments created when the angle bisector intersects the opposite side of the triangle.

1. 6 and 12 inches

2. 9 and 3 inches

3. 8 and 12 inches

4. 10 and 10 inches

Q.

In a $\Delta ABC$, AD is the bisector of $\angle A$, meeting side BC at D.
(i) If BD= 2.5 cm, AB = 5 cm and AC = 4.2 cm, find DC.
(ii) If BD= 2 cm, AB = 5 cm and DC = 3 cm, find AC.
(iii) If AB= 3.5 cm, AC = 4.2 cm and DC = 2.8 cm, find BD.
(iv) If AB =10 cm, AC =14 cm and BC = 6 cm, find BD and DC.
(v) If AC = 4.2 cm, DC = 6 cm and BC = 10 cm, find AB.
(vi) If AD= 5.6 cm, AC = 6 cm and DC = 3 cm, find BC.
(vii) If AB= 3.6 cm, BC e 6 cm and BD = 32 cm, find AC.
(viii) If AB =10 cm, AC = 6 cm and BC = 12 cm, find BD and DC

Q.

Two triangles ABC and PQR are such that AB = 3 cm, AC = 6 cm, $\angle A={70}^o,PR=9cm,\angle P={70}^{o}andPQ=4.5cm$. Show that $\Delta ABC\sim \Delta PQR$ and state the similarity criterion.

Q.

In the following figure, ABCD to a trapezium with AB||DC. If AB =9 cm, DC =18 cm, CF =13.5 cm, AP =6 cm and BE =15 cm. Calculate:

(i) BC

(ii) AF

1. BC = 20 cm, AF = 25 cm

2. BC = 22.5 cm, AF = 27 cm

3. BC = 22.5 cm, AF = 29.7 cm

4. BC = 22 cm, AF = 25 cm

Q.

In the given figure, $\angle AMN=\angle MBC={76}^o$. If a, b and c are the lengths of AM, MB and BC respectively then express the length of MN in terms of a, b and c.

Q.

ABCD is a Parallelogram is which P and Q are mid-points of apposite sides AB and CD. If AQ intersects DP at S and BQ intersects CP at R, show that
(i) APCQ is a Parallelogram
(ii) DPBQ is a parallelogram
(iii) PSQR is a parallelogram
[3 MARKS]

Q.

In the given figure, $\Delta ODC\sim \Delta OBA,\angle BOC={115}^{o}and\angle CDO={70}^o$.

Find (i) $\angle DOC$

(ii) $\angle DCO$

(iii) $\angle OAB$

(iv) $\angle OBA$.

Q.

In a right triangle ABC, right-angled at B, D is a point on hypotenuse such that $BD\sim AC$. If $DP\sim AB$ and $DQ\sim BC$ then prove that

(a) $D{Q}^2=DP.QC$

(b) $D{P}^2=DQ.AP$

Q.

In Fig. 7.145, PA, QB and RC are each perpendicular to AC. Prove that $\frac{1}{x}+\frac{1}{z}=\frac{1}{y}$

Q.

In the following figure, XY is parallel to BC AX =9 cm, XB =4.5 cm and BC =18 cm.

Then, $\frac{AY}{YC}$ = ___