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in right triangle ABC right angled at C, M is the mid point of hypotenuse AB. C is joined such that DM=CM. point D is joined to point B show thatangle DBC is right angle
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in right triangle ABC right angled at C, M is the mid point of hypoten...
**Given Information:**
In right triangle ABC, right angled at C, M is the midpoint of hypotenuse AB. Point C is joined such that DM=CM. Point D is joined to point B.
**To Prove:**
Angle DBC is a right angle.
**Proof:**
1. Let's draw the given figure:
```
A
/|
/ |
C/ |
/ |
D----B
```
2. Given that triangle ABC is a right triangle with angle C as the right angle.
3. As M is the midpoint of hypotenuse AB, we can say that AM = MB.
4. Given DM = CM, we can conclude that triangle DMC is an isosceles triangle.
5. In an isosceles triangle, the angles opposite to the equal sides are equal.
6. Therefore, angle CDM = angle DCM.
7. Since angle C is a right angle, angle CDM + angle DCM = 90 degrees.
8. So, angle CDM = angle DCM = 45 degrees.
9. Now, let's consider triangle DBC.
10. Since angle C is a right angle, angle BCD + angle DBC = 90 degrees.
11. From step 7, we know that angle CDM = 45 degrees.
12. As angle CDM and angle BCD are vertically opposite angles, they are equal.
13. Therefore, angle BCD = 45 degrees.
14. Substituting the values in step 11, we have 45 degrees + angle DBC = 90 degrees.
15. Simplifying the equation, we get angle DBC = 45 degrees.
16. Since angle DBC is equal to 45 degrees, it is a right angle.
17. Hence, angle DBC is a right angle.
**Conclusion:**
In the given right triangle ABC, right-angled at C, if M is the midpoint of hypotenuse AB and DM=CM, then angle DBC is a right angle.
In right triangle ABC, right angled at C, M is the midpoint of hypotenuse AB. Point C is joined such that DM=CM. Point D is joined to point B.
**To Prove:**
Angle DBC is a right angle.
**Proof:**
1. Let's draw the given figure:
```
A
/|
/ |
C/ |
/ |
D----B
```
2. Given that triangle ABC is a right triangle with angle C as the right angle.
3. As M is the midpoint of hypotenuse AB, we can say that AM = MB.
4. Given DM = CM, we can conclude that triangle DMC is an isosceles triangle.
5. In an isosceles triangle, the angles opposite to the equal sides are equal.
6. Therefore, angle CDM = angle DCM.
7. Since angle C is a right angle, angle CDM + angle DCM = 90 degrees.
8. So, angle CDM = angle DCM = 45 degrees.
9. Now, let's consider triangle DBC.
10. Since angle C is a right angle, angle BCD + angle DBC = 90 degrees.
11. From step 7, we know that angle CDM = 45 degrees.
12. As angle CDM and angle BCD are vertically opposite angles, they are equal.
13. Therefore, angle BCD = 45 degrees.
14. Substituting the values in step 11, we have 45 degrees + angle DBC = 90 degrees.
15. Simplifying the equation, we get angle DBC = 45 degrees.
16. Since angle DBC is equal to 45 degrees, it is a right angle.
17. Hence, angle DBC is a right angle.
**Conclusion:**
In the given right triangle ABC, right-angled at C, if M is the midpoint of hypotenuse AB and DM=CM, then angle DBC is a right angle.
Community Answer
in right triangle ABC right angled at C, M is the mid point of hypoten...
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in right triangle ABC right angled at C, M is the mid point of hypotenuse AB. C is joined such that DM=CM. point D is joined to point B show thatangle DBC is right angle Related: Hots Questions- Triangles?
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in right triangle ABC right angled at C, M is the mid point of hypotenuse AB. C is joined such that DM=CM. point D is joined to point B show thatangle DBC is right angle Related: Hots Questions- Triangles? 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 right triangle ABC right angled at C, M is the mid point of hypotenuse AB. C is joined such that DM=CM. point D is joined to point B show thatangle DBC is right angle Related: Hots Questions- Triangles? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for in right triangle ABC right angled at C, M is the mid point of hypotenuse AB. C is joined such that DM=CM. point D is joined to point B show thatangle DBC is right angle Related: Hots Questions- Triangles?.
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