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O is any point in the interior of a square ABCD such that OAB is an equilateral triangle. show that triangle OCD is an isosceles triangle. also find the measure of angle OCD .?
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O is any point in the interior of a square ABCD such that OAB is an eq...
Given:OAB is equilateral triangle. to prove:OCD is an isosceles triangle. construction:from point O, draw line from O to the points A,B,C,D. proof:line AC & BD are 2 diagonals equal. point O bisects all the diagonals. thus,AO=BO=CO=DO...CO is same as DO which proves that, OCD is isosceles.
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O is any point in the interior of a square ABCD such that OAB is an eq...
Proof that Triangle OCD is Isosceles:
- Since OAB is an equilateral triangle, all sides are equal in length. Therefore, OA = AB = OB.
- Since O is the midpoint of AB, it divides the square diagonally into two equal parts. So, AO = OB.
- In triangle AOC, AO = OC (as O is the midpoint of AC in the square).
- Combining the above two points, we have AO = OC = OB. Therefore, triangle OCD is isosceles.
Finding the Measure of Angle OCD:
- Since triangle OCD is isosceles, OC = OD. Therefore, angles OCD and ODC are equal as they are base angles of an isosceles triangle.
- The sum of angles in a triangle is 180 degrees. So, in triangle OCD, we have: OCD + ODC + O = 180 degrees.
- Since OAB is an equilateral triangle, angle AOB is 60 degrees.
- Since square ABCD is a 90-degree angle, angle AOC is 90 degrees.
- Substituting these values into the equation from the previous point, we get: OCD + 60 + 90 = 180 degrees.
- Solving for OCD, we find: OCD = 30 degrees.
Therefore, triangle OCD is an isosceles triangle with angle OCD measuring 30 degrees.
- Since OAB is an equilateral triangle, all sides are equal in length. Therefore, OA = AB = OB.
- Since O is the midpoint of AB, it divides the square diagonally into two equal parts. So, AO = OB.
- In triangle AOC, AO = OC (as O is the midpoint of AC in the square).
- Combining the above two points, we have AO = OC = OB. Therefore, triangle OCD is isosceles.
Finding the Measure of Angle OCD:
- Since triangle OCD is isosceles, OC = OD. Therefore, angles OCD and ODC are equal as they are base angles of an isosceles triangle.
- The sum of angles in a triangle is 180 degrees. So, in triangle OCD, we have: OCD + ODC + O = 180 degrees.
- Since OAB is an equilateral triangle, angle AOB is 60 degrees.
- Since square ABCD is a 90-degree angle, angle AOC is 90 degrees.
- Substituting these values into the equation from the previous point, we get: OCD + 60 + 90 = 180 degrees.
- Solving for OCD, we find: OCD = 30 degrees.
Therefore, triangle OCD is an isosceles triangle with angle OCD measuring 30 degrees.
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O is any point in the interior of a square ABCD such that OAB is an equilateral triangle. show that triangle OCD is an isosceles triangle. also find the measure of angle OCD .?
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O is any point in the interior of a square ABCD such that OAB is an equilateral triangle. show that triangle OCD is an isosceles triangle. also find the measure of angle OCD .? 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 O is any point in the interior of a square ABCD such that OAB is an equilateral triangle. show that triangle OCD is an isosceles triangle. also find the measure of angle OCD .? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for O is any point in the interior of a square ABCD such that OAB is an equilateral triangle. show that triangle OCD is an isosceles triangle. also find the measure of angle OCD .?.
O is any point in the interior of a square ABCD such that OAB is an equilateral triangle. show that triangle OCD is an isosceles triangle. also find the measure of angle OCD .? 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 O is any point in the interior of a square ABCD such that OAB is an equilateral triangle. show that triangle OCD is an isosceles triangle. also find the measure of angle OCD .? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for O is any point in the interior of a square ABCD such that OAB is an equilateral triangle. show that triangle OCD is an isosceles triangle. also find the measure of angle OCD .?.
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