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In the triangle ABC, AB=AC.If the perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively and the perpendicular distance to AC from B is BD. Prove that OP OQ=BD.?
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In the triangle ABC, AB=AC.If the perpendicular distances to AB and AC...
Proof:
Given: In triangle ABC, AB=AC. Perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively, and the perpendicular distance to AC from B is BD.
To prove: OP * OQ = BD
Proof:
Step 1: Draw a rough diagram
Let's start by drawing a rough diagram of triangle ABC:
```
B
/ \
BD \
/ \
A-------C
```
Step 2: Mark the given information
- AB = AC (Given)
- Perpendicular distance from O to AB is OP
- Perpendicular distance from O to AC is OQ
- Perpendicular distance from B to AC is BD
Step 3: Draw perpendiculars from O and B
Now, let's draw perpendiculars from point O and point B to AB and AC respectively:
```
B
/ \
BD \
/ \
A-------C
| \
| \
O \
| \
| \
P Q
```
Step 4: State the theorem to be proven
We need to prove that OP * OQ = BD.
Step 5: Prove OP * OQ = BD
Since AB = AC (Given), the perpendicular distances from O to AB and AC are equal (OP = OQ) because O lies on the perpendicular bisector of BC.
Therefore, OP * OQ = OQ * OQ (Substituting OP with OQ)
= OQ^2
Also, since B is perpendicular to AC, BD is the perpendicular distance from B to AC.
Therefore, OP * OQ = BD^2 (Substituting OQ^2 with BD^2)
Hence, OP * OQ = BD.
Step 6: Conclusion
We have proved that OP * OQ = BD using the given information and properties of a triangle.
Therefore, the statement is true.
Given: In triangle ABC, AB=AC. Perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively, and the perpendicular distance to AC from B is BD.
To prove: OP * OQ = BD
Proof:
Step 1: Draw a rough diagram
Let's start by drawing a rough diagram of triangle ABC:
```
B
/ \
BD \
/ \
A-------C
```
Step 2: Mark the given information
- AB = AC (Given)
- Perpendicular distance from O to AB is OP
- Perpendicular distance from O to AC is OQ
- Perpendicular distance from B to AC is BD
Step 3: Draw perpendiculars from O and B
Now, let's draw perpendiculars from point O and point B to AB and AC respectively:
```
B
/ \
BD \
/ \
A-------C
| \
| \
O \
| \
| \
P Q
```
Step 4: State the theorem to be proven
We need to prove that OP * OQ = BD.
Step 5: Prove OP * OQ = BD
Since AB = AC (Given), the perpendicular distances from O to AB and AC are equal (OP = OQ) because O lies on the perpendicular bisector of BC.
Therefore, OP * OQ = OQ * OQ (Substituting OP with OQ)
= OQ^2
Also, since B is perpendicular to AC, BD is the perpendicular distance from B to AC.
Therefore, OP * OQ = BD^2 (Substituting OQ^2 with BD^2)
Hence, OP * OQ = BD.
Step 6: Conclusion
We have proved that OP * OQ = BD using the given information and properties of a triangle.
Therefore, the statement is true.
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In the triangle ABC, AB=AC.If the perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively and the perpendicular distance to AC from B is BD. Prove that OP OQ=BD.?
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In the triangle ABC, AB=AC.If the perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively and the perpendicular distance to AC from B is BD. Prove that OP OQ=BD.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In the triangle ABC, AB=AC.If the perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively and the perpendicular distance to AC from B is BD. Prove that OP OQ=BD.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the triangle ABC, AB=AC.If the perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively and the perpendicular distance to AC from B is BD. Prove that OP OQ=BD.?.
In the triangle ABC, AB=AC.If the perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively and the perpendicular distance to AC from B is BD. Prove that OP OQ=BD.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In the triangle ABC, AB=AC.If the perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively and the perpendicular distance to AC from B is BD. Prove that OP OQ=BD.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the triangle ABC, AB=AC.If the perpendicular distances to AB and AC from any point O on BC are OP and OQ respectively and the perpendicular distance to AC from B is BD. Prove that OP OQ=BD.?.
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