Find the best tutors and institutes for Class 10 Tuition
Q6:
In Fig. 6.21, $A$, $B$ and $C$ are points on $OP$, $OQ$ and $OR$ respectively such that $AB \parallel PQ$ and $AC \parallel PR$. Show that $BC \parallel QR$.
In Fig. 6.21, $A$, $B$ and $C$ are points on $OP$, $OQ$ and $OR$ respectively such that $AB \parallel PQ$ and $AC \parallel PR$. Show that $BC \parallel QR$.
Solution :
Given:
In $\triangle OPQ$, $A$ is on $OP$ and $B$ is on $OQ$ such that $AB \parallel PQ$.
In $\triangle OPR$, $A$ is on $OP$ and $C$ is on $OR$ such that $AC \parallel PR$.
To Prove:
$BC \parallel QR$.
Step 1: Applying Thales Theorem (Basic Proportionality Theorem) in $\triangle OPQ$
According to the Basic Proportionality Theorem (BPT), if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Since $AB \parallel PQ$ in $\triangle OPQ$:
$\frac{OA}{AP} = \frac{OB}{BQ}$ --- (Equation 1) [By BPT]
Step 2: Applying Thales Theorem in $\triangle OPR$
Since $AC \parallel PR$ in $\triangle OPR$:
$\frac{OA}{AP} = \frac{OC}{CR}$ --- (Equation 2) [By BPT]
Step 3: Comparing the Equations
From Equation 1 and Equation 2, we observe that the left-hand sides are identical ($\frac{OA}{AP}$). Therefore, we can equate the right-hand sides:
$\frac{OB}{BQ} = \frac{OC}{CR}$ --- (Equation 3)
Step 4: Applying the Converse of the Basic Proportionality Theorem
The Converse of the Basic Proportionality Theorem states that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
In $\triangle OQR$, we have established from Equation 3 that:
$\frac{OB}{BQ} = \frac{OC}{CR}$
Therefore, by the Converse of the Basic Proportionality Theorem, it follows that:
$BC \parallel QR$
Final Answer: Since the ratios of the segments on sides $OQ$ and $OR$ are equal ($\frac{OB}{BQ} = \frac{OC}{CR}$), by the Converse of the Basic Proportionality Theorem, $BC \parallel QR$ is proved.
More Questions from Class 10 Mathematics Triangles EXERCISE 6.2
- Q1(i): In Fig. 6.17, (i) and (ii), $DE \parallel BC$. Find $EC$ in (i).
- Q1(ii): In Fig. 6.17, (i) and (ii), $DE \parallel BC$. Find $AD$ in (ii).
- Q10: The diagonals of a quadrilateral $ABCD$ intersect each other at the point $O$ such that $\frac{AO}{BO} = \frac{CO}{DO}$. Show that $ABCD$ is a trapezium.
- Q2(i): $E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\triangle PQR$. For each of the following cases, state whether $EF \parallel QR$ : (i) $PE = 3.9$ cm, $EQ = 3$ cm, $PF = 3.6$ cm and $FR = 2.4$ cm
- Q2(ii): $E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\triangle PQR$. For each of the following cases, state whether $EF \parallel QR$ : (ii) $PE = 4$ cm, $QE = 4.5$ cm, $PF = 8$ cm and $RF = 9$ cm
- Q2(iii): $E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\triangle PQR$. For each of the following cases, state whether $EF \parallel QR$ : (iii) $PQ = 1.28$ cm, $PR = 2.56$ cm, $PE = 0.18$ cm and $PF = 0.36$ cm
- Q3: In Fig. 6.18, if $LM \parallel CB$ and $LN \parallel CD$, prove that $\frac{AM}{AB} = \frac{AN}{AD}$.
- Q4: In Fig. 6.19, $DE \parallel AC$ and $DF \parallel AE$. Prove that $\frac{BF}{FE} = \frac{BE}{EC}$.
- Q5: In Fig. 6.20, $DE \parallel OQ$ and $DF \parallel OR$. Show that $EF \parallel QR$.
- Q7: 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).
- Q8: 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).
- Q9: $ABCD$ is a trapezium in which $AB \parallel DC$ and its diagonals intersect each other at the point $O$. Show that $\frac{AO}{BO} = \frac{CO}{DO}$.
CBSE Solutions for Class 10 Mathematics Triangles
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Triangles
Top Tutor
Mira Road, Mira-Bhayandar
10 yrs of
Exp
7 students
84 hrs of classes completed
600
per hour
Student's Home
Tutor's home
Math and physics teacher. Expert in cbse education and assessment patten. Been an cbse examiner for 5 continuous years. Aware of the curriculum of ssc, icse and cbse grade 7-10 math and physics. Tutoring since 2006 and been quite successful in giving results.
I would like to say the class was really helpful. I was able to catch a hold on various topics taught in class. Sir also has a very kind nature and he helped us in various way. Overall I loved the class.
Top Tutor
Sector 19, Noida
3 yrs of
Exp
12 students
578 hrs of classes completed
Online Classes
• 3+ years of teaching experience. • teaching online tuitions of Mathematics for Class 10-12th. • pursued M.Sc. Mathematics from University of Delhi & B.Sc. Mathematical Sciences from University of Delhi. Passionate about solving mathematical problems and I have helped many students to overcome the fear of Maths.
Im so grateful that i got in contact with palak ma'am in my difficult phase.. my boards are on the end and she is standing with me to support me and help me till the end..thanks alot will make her proud for sure !! amazing teacher !! grateful
Top Tutor
Uttorayon Township, Siliguri
3 yrs of
Exp
3 students
91 hrs of classes completed
350
per hour
Online Classes
Student's Home
Tutor's home
Ability to connect with children (or with older students), on a personal level. communication and social skills.Teaching can be quite satisfying for people who do it well. I know this because I am a teacher, too. Although I do not think I'm always the best teacher, do I know that the majority of my students love me, and I guess this is partly why good things happen in my classroom. I have spent a lot of time thinking about and done a lot of research in to the question of what characteristics make a teacher effective and how I can succeed in the classroom. Although every great teacher has their own special, unique style, I have found that there are nine specific universal qualities that are necessary for anyone who wants to teach effectively, whether they want to teach in an elementary school or a university. If any teacher possesses the following characteristics and qualities, he/she can become a very good teacher with large fan base.
Top Tutor
Burul Rd Burul, Burul
5 yrs of
Exp
100
per hour
Online Classes
Student's Home
Tutor's home
I am a teacher of math. I am giving home/online/tutor home tuition since 2016. My key skills are computer language. I am b. Tech graduate. I can teach short tricks that can help students in their future life.
Top Tutor
Mahmoorganj Tulsipur, Varanasi
4 yrs of
Exp
12 students
627 hrs of classes completed
100
per hour
Online Classes
Teaching methodology is such that even weakest students can understand the concept very easily through real life analogy
Great teacher i understand subjects with better clarity now and he teaches great and it sure helped me alot
Find more Tutor for Triangles in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
