Class 10 › Mathematics Triangles Class 10 Mathematics · Chapter 6
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View all 15 amazing Marks Booster kits → This chapter studies similarity of triangles — figures of the same shape but not necessarily the same size. Two triangles are similar when their corresponding angles are equal and corresponding sides are in proportion. The Basic Proportionality Theorem (Thales’ theorem) states that a line parallel to one side of a triangle divides the other two sides in the same ratio. Criteria such as AA, SSS and SAS establish similarity, and similar triangles are used to prove results and find unknown lengths in geometric and real-life problems involving heights and shadows.
Similar triangles
Triangles with equal corresponding angles and proportional sides.
Basic Proportionality Theorem
A line parallel to one side divides the other two sides proportionally.
AA criterion
Two triangles are similar if two pairs of angles are equal.
Corresponding sides
Sides opposite to equal angles in similar triangles.
Q1 State the Basic Proportionality Theorem (Thales’ theorem). Answer Q2 State the AA similarity criterion. Answer Q3 When are two triangles said to be similar? Answer Q4 State the Basic Proportionality Theorem (Thales’ theorem). Answer Q5 Name the criteria used to prove two triangles similar. Answer Q6 State the AA similarity criterion. Answer Q7 How are the areas of two similar triangles related to their sides? Answer Q8 In a triangle, a line parallel to BC meets AB at D and AC at E. If \(AD/DB = 3/4\), find \(AE/EC\). Answer Q9 If two similar triangles have corresponding sides in the ratio \(2 : 3\), what is the ratio of their areas? Answer Q10 Give one real-life use of similar triangles. Answer Q1 1. Fill in the blanks using the correct word given in brackets :
(i) All circles are ______. (congruent, similar)
(ii) All squares are ______. (similar, congruent)
(iii) All ______ triangles are similar. (isosceles, equilateral)
(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are ______ and (b) their corresponding sides are ______. (equal, proportional) Answer Q2 2. Give two different examples of pair of
(i) similar figures. (ii) non-similar figures. Answer Q3 3. State whether the following quadrilaterals are similar or not: Answer Q1 1. In Fig. 6.17, (i) and (ii), DE
parallelBC. Find EC in (i) and AD in (ii). Answer Q2 2. E and F are points on the sides PQ and PR respectively of a
trianglePQR. For each of the following cases, state whether EF
parallelQR :
(i) PE=3.9
textcm,EQ=3
textcm,PF=3.6
textcm and FR=2.4
textcm
(ii) PE=4
textcm,QE=4.5
textcm,PF=8
textcm and RF=9
textcm
(iii) PQ=1.28
textcm,PR=2.56
textcm,PE=0.18
textcm and PF=0.36
textcm Answer Q3 3. In Fig. 6.18, if LM
parallelCB and LN
parallelCD, prove that
\[\n\frac{AM}{AB}=\frac{AN}{AD}.\n\] Answer Q4 4. In Fig. 6.19, DE
parallelAC and DF
parallelAE. Prove that
\[\n\frac{BF}{FE}=\frac{BE}{EC}.\n\] Answer Q5 5. In Fig. 6.20, DE
parallelOQ and DF
parallelOR. Show that EF
parallelQR. Answer Q6 6. In Fig. 6.21, A,B and C are points on OP,OQ and OR respectively such that AB
parallelPQ and AC
parallelPR. Show that BC
parallelQR. Answer Q7 7. Using Theorem 6.1, 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 you have proved it in Class IX). Answer Q8 8. Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX). Answer Q9 9. ABCD is a trapezium in which AB
parallelDC and its diagonals intersect each other at the point O. Show that
\[\n\frac{AO}{BO}=\frac{CO}{DO}.\n\] Answer Q10 10. The diagonals of a quadrilateral ABCD intersect each other at the point O such that
\[\n\frac{AO}{BO}=\frac{CO}{DO}.\n\]
Show that ABCD is a trapezium. Answer Q1 1. State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form : Answer Q2 2. In Fig. 6.35,
triangleODC
sim
triangleOBA,
angleBOC=125
∘
and
angleCDO=70
∘
. Find
angleDOC,
angleDCO and
angleOAB. Answer Q3 3. Diagonals AC and BD of a trapezium ABCD with AB
parallelDC intersect each other at the point O. Using a similarity criterion for two triangles, show that
\[\n\frac{OA}{OC}=\frac{OB}{OD}.\n\] Answer Q4 4. In Fig. 6.36,
\[\n\frac{QR}{QS}=\frac{QT}{PR}\n\]
and
angle1=
angle2. Show that
trianglePQS
sim
triangleTQR. Answer Q5 5. S and T are points on sides PR and QR of
trianglePQR such that
angleP=
angleRTS. Show that
triangleRPQ
sim
triangleRTS. Answer Q6 6. In Fig. 6.37, if
triangleABE
cong
triangleACD, show that
triangleADE
sim
triangleABC. Answer Q7 7. In Fig. 6.38, altitudes AD and CE of
triangleABC intersect each other at the point P. Show that:
(i)
triangleAEP
sim
triangleCDP
(ii)
triangleABD
sim
triangleCBE
(iii)
triangleAEP
sim
triangleADB
(iv)
trianglePDC
sim
triangleBEC Answer Q8 8. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that
triangleABE
sim
triangleCFB. Answer Q9 9. In Fig. 6.39, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that:
(i)
triangleABC
sim
triangleAMP
(ii)
\[\n\frac{CA}{PA}=\frac{BC}{MP}.\n\] Answer Q10 10. CD and GH are respectively the bisectors of
angleACB and
angleEGF such that D and H lie on sides AB and FE of
triangleABC and
triangleEFG respectively. If
triangleABC
sim
triangleFEG, show that:
(i)
\[\n\frac{CD}{GH}=\frac{AC}{FG}\n\]
(ii)
triangleDCB
sim
triangleHGE
(iii)
triangleDCA
sim
triangleHGF Answer Q11 11. In Fig. 6.40, E is a point on side CB produced of an isosceles triangle ABC with AB=AC. If AD
perpBC and EF
perpAC, prove that
triangleABD
sim
triangleECF. Answer Q12 12. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of
trianglePQR (see Fig. 6.41). Show that
triangleABC
sim
trianglePQR. Answer Q13 13. D is a point on the side BC of a triangle ABC such that
angleADC=
angleBAC. Show that CA
2
=CB
cdotCD. Answer Q14 14. 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
triangleABC
sim
trianglePQR. Answer Q15 15. A vertical pole of length 6
textm casts a shadow 4
textm long on the ground and at the same time a tower casts a shadow 28
textm long. Find the height of the tower. Answer Q16 16. If AD and PM are medians of triangles ABC and PQR, respectively where
triangleABC
sim
trianglePQR, prove that
\[\n\frac{AB}{PQ}=\frac{AD}{PM}.\n\] Answer
Two figures having the same shape but not necessarily the same size are called:
A similar B congruent C equal D identical
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