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The side BC of a triangle ABC is bisected at D;O is any point in AD.BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid point of OX. Prove that AO:AX = AF:AB and show that FE || BC?
Verified Answer
The side BC of a triangle ABC is bisected at D;O is any point in AD.BO...
Given : The side BC of a triangle ABC is bisected at D; 0 is any point in AD.
BO ,CO produced meet AC, AB in E , F respectively, and AD is produced to X
so that D is the mid point of oX.
To Prove : AO : AX = AF : AB and show that EF is parallel to BC.
Construction: Join BX and CX.
Proof: In quadrilateral BOCX,BD = DC and DO = DX (given)
BOCX is a parallelogram (When the diagonals of a quadrilateral bisect
each other, then the quad. is a parallelogram)
∴BX || CO (Definition of a parallelogram)
or BX || FO.
In △ABX, BX || FO(proved)
∴AO : AX = AF : AB (using B.P.T)....(i)
Similarly, AO : AX = AE : AC....(ii)
From (i) and (ii),
AF : AB = AE : AC
By corollary to B.P.T, EF is parallel to BC.
This question is part of UPSC exam. View all Class 10 courses
Most Upvoted Answer
The side BC of a triangle ABC is bisected at D;O is any point in AD.BO...
Given:
- Triangle ABC
- Side BC is bisected at D
- Point O is any point on AD
- BO and CO produced meet AC and AB at E and F respectively
- AD is extended to X such that D is the midpoint of OX
We are required to prove two things:
1. AO:AX = AF:AB
2. FE || BC
Proof:
1. Proving AO:AX = AF:AB:
Let's consider triangle ABC.
Using the Midpoint Theorem:
Since D is the midpoint of BC, we can say that AD is parallel to EF.
Using the Intercept Theorem:
Since BO and CO produced meet AC and AB at E and F respectively, we can say that AF:FB = AE:EC.
Now, let's consider triangle AOX.
Using the Midpoint Theorem:
Since D is the midpoint of OX, we can say that AD is parallel to FE.
Using the Intercept Theorem:
Since BO and CO produced meet AC and AB at E and F respectively, we can say that AF:FB = AE:EC.
From the above two conclusions, we can say that AO:AX = AF:AB.
2. Proving FE || BC:
Since AD is parallel to FE and BC, and AD intersects BC at D, we can say that FE || BC.
Therefore, we have proved that FE || BC.
To summarize:
- AO:AX = AF:AB
- FE || BC
- Triangle ABC
- Side BC is bisected at D
- Point O is any point on AD
- BO and CO produced meet AC and AB at E and F respectively
- AD is extended to X such that D is the midpoint of OX
We are required to prove two things:
1. AO:AX = AF:AB
2. FE || BC
Proof:
1. Proving AO:AX = AF:AB:
Let's consider triangle ABC.
Using the Midpoint Theorem:
Since D is the midpoint of BC, we can say that AD is parallel to EF.
Using the Intercept Theorem:
Since BO and CO produced meet AC and AB at E and F respectively, we can say that AF:FB = AE:EC.
Now, let's consider triangle AOX.
Using the Midpoint Theorem:
Since D is the midpoint of OX, we can say that AD is parallel to FE.
Using the Intercept Theorem:
Since BO and CO produced meet AC and AB at E and F respectively, we can say that AF:FB = AE:EC.
From the above two conclusions, we can say that AO:AX = AF:AB.
2. Proving FE || BC:
Since AD is parallel to FE and BC, and AD intersects BC at D, we can say that FE || BC.
Therefore, we have proved that FE || BC.
To summarize:
- AO:AX = AF:AB
- FE || BC
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The side BC of a triangle ABC is bisected at D;O is any point in AD.BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid point of OX. Prove that AO:AX = AF:AB and show that FE || BC?
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The side BC of a triangle ABC is bisected at D;O is any point in AD.BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid point of OX. Prove that AO:AX = AF:AB and show that FE || BC? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The side BC of a triangle ABC is bisected at D;O is any point in AD.BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid point of OX. Prove that AO:AX = AF:AB and show that FE || BC? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The side BC of a triangle ABC is bisected at D;O is any point in AD.BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid point of OX. Prove that AO:AX = AF:AB and show that FE || BC?.
The side BC of a triangle ABC is bisected at D;O is any point in AD.BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid point of OX. Prove that AO:AX = AF:AB and show that FE || BC? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The side BC of a triangle ABC is bisected at D;O is any point in AD.BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid point of OX. Prove that AO:AX = AF:AB and show that FE || BC? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The side BC of a triangle ABC is bisected at D;O is any point in AD.BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid point of OX. Prove that AO:AX = AF:AB and show that FE || BC?.
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