Find the best tutors and institutes for Class 10 Tuition
Q13:
$D$ is a point on the side $BC$ of a triangle $ABC$ such that $\angle ADC = \angle BAC$. Show that $CA^2 = CB \cdot CD$.
Solution :
Given: A triangle $ABC$ and a point $D$ on the side $BC$ such that $\angle ADC = \angle BAC$.
To Prove: $CA^2 = CB \cdot CD$.
Step 1: Identifying the Triangles to be Compared
To prove the relationship $CA^2 = CB \cdot CD$, we observe that this can be rewritten as the ratio $\frac{CA}{CB} = \frac{CD}{CA}$. This suggests that we should consider the two triangles $\triangle ADC$ and $\triangle BAC$.
Step 2: Establishing Similarity Criteria
We compare $\triangle ADC$ and $\triangle BAC$ based on the following observations:
1. In $\triangle ADC$ and $\triangle BAC$, we are given that $\angle ADC = \angle BAC$. [Given in the problem statement]
2. In $\triangle ADC$ and $\triangle BAC$, the angle $\angle C$ is common to both triangles. That is, $\angle ACD = \angle BCA$. [Common angle]
Step 3: Applying the AA (Angle-Angle) Similarity Criterion
Since two angles of $\triangle ADC$ are equal to two corresponding angles of $\triangle BAC$, by the AA similarity criterion, the triangles are similar.
Therefore, $\triangle ADC \sim \triangle BAC$. [By AA Similarity Criterion]
Step 4: Utilizing the Properties of Similar Triangles
When two triangles are similar, the ratios of their corresponding sides are equal. Based on the correspondence established in Step 3:
$\frac{AD}{BA} = \frac{DC}{AC} = \frac{AC}{BC}$
[Since corresponding sides of similar triangles are proportional]
Step 5: Deriving the Final Equation
We take the relevant parts of the proportionality ratio from Step 4:
$\frac{DC}{AC} = \frac{AC}{BC}$
By performing cross-multiplication:
$AC \cdot AC = DC \cdot BC$
$AC^2 = BC \cdot DC$
Since $AC$ is the same as $CA$ and $BC$ is the same as $CB$, we can rewrite this as:
$CA^2 = CB \cdot CD$
Final Answer: Hence, it is proved that $CA^2 = CB \cdot CD$.
More Questions from Class 10 Mathematics Triangles EXERCISE 6.3
- Q1(i): 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 : (i)
- Q1(ii): 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 : (ii)
- Q1(iii): 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 : (iii)
- Q1(iv): 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 : (iv)
- Q1(v): 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 : (v)
- Q1(vi): 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 : (vi)
- Q10(i): $CD$ and $GH$ are respectively the bisectors of $\angle ACB$ and $\angle EGF$ such that $D$ and $H$ lie on sides $AB$ and $FE$ of $\triangle ABC$ and $\triangle EFG$ respectively. If $\triangle ABC \sim \triangle FEG$, show that: (i) $\frac{CD}{GH} = \frac{AC}{FG}$
- Q10(ii): $CD$ and $GH$ are respectively the bisectors of $\angle ACB$ and $\angle EGF$ such that $D$ and $H$ lie on sides $AB$ and $FE$ of $\triangle ABC$ and $\triangle EFG$ respectively. If $\triangle ABC \sim \triangle FEG$, show that: (ii) $\triangle DCB \sim \triangle HGE$
- Q10(iii): $CD$ and $GH$ are respectively the bisectors of $\angle ACB$ and $\angle EGF$ such that $D$ and $H$ lie on sides $AB$ and $FE$ of $\triangle ABC$ and $\triangle EFG$ respectively. If $\triangle ABC \sim \triangle FEG$, show that: (iii) $\triangle DCA \sim \triangle HGF$
- Q11: In Fig. 6.40, $E$ is a point on side $CB$ produced of an isosceles triangle $ABC$ with $AB = AC$. If $AD \perp BC$ and $EF \perp AC$, prove that $\triangle ABD \sim \triangle ECF$.
- Q12: Sides $AB$ and $BC$ and median $AD$ of a triangle $ABC$ are respectively proportional to sides $PQ$ and $QR$ and median $PM$ of $\triangle PQR$ (see Fig. 6.41). Show that $\triangle ABC \sim \triangle PQR$.
- Q14: 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 $\triangle ABC \sim \triangle PQR$.
- Q15: 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 tower.
- Q16: If $AD$ and $PM$ are medians of triangles $ABC$ and $PQR$, respectively where $\triangle ABC \sim \triangle PQR$, prove that $\frac{AB}{PQ} = \frac{AD}{PM}$.
- Q2: In Fig. 6.35, $\triangle ODC \sim \triangle OBA$, $\angle BOC = 125^{\circ}$ and $\angle CDO = 70^{\circ}$. Find $\angle DOC$, $\angle DCO$ and $\angle OAB$.
- Q3: Diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB \parallel DC$ intersect each other at the point $O$. Using a similarity criterion for two triangles, show that $\frac{OA}{OC} = \frac{OB}{OD}$.
- Q4: In Fig. 6.36, $\frac{QR}{QS} = \frac{QT}{PR}$ and $\angle 1 = \angle 2$. Show that $\triangle PQS \sim \triangle TQR$.
- Q5: $S$ and $T$ are points on sides $PR$ and $QR$ of $\triangle PQR$ such that $\angle P = \angle RTS$. Show that $\triangle RPQ \sim \triangle RTS$.
- Q6: In Fig. 6.37, if $\triangle ABE \cong \triangle ACD$, show that $\triangle ADE \sim \triangle ABC$.
- Q7(i): In Fig. 6.38, altitudes $AD$ and $CE$ of $\triangle ABC$ intersect each other at the point $P$. Show that: (i) $\triangle AEP \sim \triangle CDP$
- Q7(ii): In Fig. 6.38, altitudes $AD$ and $CE$ of $\triangle ABC$ intersect each other at the point $P$. Show that: (ii) $\triangle ABD \sim \triangle CBE$
- Q7(iii): In Fig. 6.38, altitudes $AD$ and $CE$ of $\triangle ABC$ intersect each other at the point $P$. Show that: (iii) $\triangle AEP \sim \triangle ADB$
- Q7(iv): In Fig. 6.38, altitudes $AD$ and $CE$ of $\triangle ABC$ intersect each other at the point $P$. Show that: (iv) $\triangle PDC \sim \triangle BEC$
- Q8: $E$ is a point on the side $AD$ produced of a parallelogram $ABCD$ and $BE$ intersects $CD$ at $F$. Show that $\triangle ABE \sim \triangle CFB$.
- Q9(i): In Fig. 6.39, $ABC$ and $AMP$ are two right triangles, right angled at $B$ and $M$ respectively. Prove that: (i) $\triangle ABC \sim \triangle AMP$
- Q9(ii): In Fig. 6.39, $ABC$ and $AMP$ are two right triangles, right angled at $B$ and $M$ respectively. Prove that: (ii) $\frac{CA}{PA} = \frac{BC}{MP}$
CBSE Solutions for Class 10 Mathematics Triangles
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Triangles
Super Tutor
T Nagar, Chennai
10 yrs of
Exp
43 students
1069 hrs of classes completed
500
per hour
Online Classes
Teaching Experience in Detail (Class 10 Tuition) I have over 10 years of experience teaching CBSE, ICSE, and State Board Class 10 students, helping them achieve excellent results in their board examinations through a structured and concept-oriented teaching approach. I specialize in Science, Mathematics, Social Science, and English, ensuring students develop a thorough understanding of each subject rather than relying on rote learning. My teaching methodology begins with strengthening fundamental concepts, followed by chapter-wise practice, NCERT-based learning, important board questions, competency-based questions, case study questions, and previous years' board papers. Regular assignments, periodic tests, and detailed performance analysis help students identify weak areas and improve consistently. I provide personalized attention to every student by adapting my teaching style to their learning pace. My classes include interactive discussions, doubt-clearing sessions, revision plans, and exam-oriented strategies that build confidence and improve time management. My objective is to help students not only score high marks in the Class 10 Board Examination but also develop strong analytical and problem-solving skills that prepare them for higher secondary education. Through consistent guidance, disciplined practice, and individual mentoring, I strive to make learning enjoyable, effective, and result-oriented.
Dr Nandhini ma'am's methodology is appreciable, and she is approachable too. Her repeated questioning of the kids made them feel lively in class, and it helped a lot. Thank you, ma'am, for your kind, encouraging words to my kid.
Super Tutor
Nerul Sector 20, Thane
10 yrs of
Exp
62 students
2839 hrs of classes completed
1200
per hour
Online Classes
I have done b.Tech in electrical engineering from IIT KANPUR. I taught one year in akash institute patna, 2 years in fiitjee patna. Currently I am teaching in unacademy. I have taught more than 6000 jee and neet aspirants. In which more than 500 students are selected in iitand aiims. If some one is interested to improve their physics upto jee advanced level take help with me .And see the magic. I can help students in their all type of assignment from any institute like allen, fiitjee, narayna, akash, resonance. I can help students to solve both volume of h.C verma within two month if some one is intrested in solving I.E.Irodov join me and see the next level of physics. My way of teaching is so easy and so advanced that student will realyy enjoy it. Don't waste your time in making decision just join and experience my level
He is a very dedicated teacher with excellent knowledge on topic. He knows Subject very well and teach accordingly to kids.
Top Tutor
Sector 150, Noida
5 yrs of
Exp
800
per hour
Student's Home
Tutor's home
Having 2 years experience in teaching class 6th to 10th across different boards.
Nanmangalam, Chennai
3 yrs of
Exp
2 students
88 hrs of classes completed
260
per hour
Online Classes
Tutor's home
Hi, my name is M.Praveshika. I am a graduate in Master of Philosophy (Mathematics) from Stella Maris College. During my degree, I have developed an easy algorithm for solving certain graphs in my specialization paper. I spent my spare time by tutoring my cousins and working with them gave me a new understanding how students are facing problems in understanding the concepts. So i started my career as a Subject Matter Expert (EzeeScore) which enabled me to utilize my skills and knowledge for the growth of the students as well as to the organisation. My key skills are Leadership, Time maganement, Decision making and Problem solving.
Top Tutor
Electronic City Electronic City Phase II, Bangalore
5 yrs of
Exp
4 students
38 hrs of classes completed
150
per hour
Online Classes
Tutor's home
I was completed my m.Tech in engineering. I was maintained tuition center from 7 years in my home to all subjects from 1st grade to 10th grades across different boards including CBSE,ICSE,STATE BOARD (expert in math and physics) and I have 2 years experience in engineering college and i took classes for engineering students . My teaching style is create every student in to think and learn and convert every student in to passionate about their subjects. Over the years i helped to so many students to grown their career and overcome their fear about maths ,physics.
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
