To show that OA = OB = OC in a right-angled triangle ABC with right angle at B, where O is the midpoint of the hypotenuse AC, you can use the Pythagorean Theorem.
Let's denote the length of the hypotenuse AC as c. Then, OA = OC = c/2, because O is the midpoint of AC.
Now, to show that OB = c/2, we can use the Pythagorean Theorem. In triangle ABC, we have:
AB^2 + BC^2 = AC^2
Since it's a right-angled triangle, AB^2 + BC^2 = c^2.
Since O is the midpoint of AC, we can say that AO = CO = c/2.
Now, we have a right-angled triangle OBC where OB is the hypotenuse, and BC is half of the hypotenuse AC (c/2). Applying the Pythagorean Theorem in this smaller triangle:
OB^2 = OC^2 + BC^2
OB^2 = (c/2)^2 + (c/2)^2
OB^2 = c^2/4 + c^2/4
OB^2 = c^2/2
Taking the square root of both sides:
OB = c/√2
Now, we know that OA = OC = c/2 and OB = c/√2. Since c/√2 = (c/2) * √2, we can see that OB = OA = OC, and we have successfully shown that OA = OB = OC in the given triangle.
Answer:To show that OA = OB = OC in a right-angled triangle ABC with right angle at B, where O is the midpoint of the hypotenuse AC, you can use the Pythagorean Theorem.
Let's denote the length of the hypotenuse AC as c. Then, OA = OC = c/2, because O is the midpoint of AC.
Now, to show that OB = c/2, we can use the Pythagorean Theorem. In triangle ABC, we have:
AB^2 + BC^2 = AC^2
Since it's a right-angled triangle, AB^2 + BC^2 = c^2.
Since O is the midpoint of AC, we can say that AO = CO = c/2.
Now, we have a right-angled triangle OBC where OB is the hypotenuse, and BC is half of the hypotenuse AC (c/2). Applying the Pythagorean Theorem in this smaller triangle:
OB^2 = OC^2 + BC^2
OB^2 = (c/2)^2 + (c/2)^2
OB^2 = c^2/4 + c^2/4
OB^2 = c^2/2
Taking the square root of both sides:
OB = c/√2
Now, we know that OA = OC = c/2 and OB = c/√2. Since c/√2 = (c/2) * √2, we can see that OB = OA = OC, and we have successfully shown that OA = OB = OC in the given triangle.
Answers & Comments
Answer:
To show that OA = OB = OC in a right-angled triangle ABC with right angle at B, where O is the midpoint of the hypotenuse AC, you can use the Pythagorean Theorem.
Let's denote the length of the hypotenuse AC as c. Then, OA = OC = c/2, because O is the midpoint of AC.
Now, to show that OB = c/2, we can use the Pythagorean Theorem. In triangle ABC, we have:
AB^2 + BC^2 = AC^2
Since it's a right-angled triangle, AB^2 + BC^2 = c^2.
Since O is the midpoint of AC, we can say that AO = CO = c/2.
Now, we have a right-angled triangle OBC where OB is the hypotenuse, and BC is half of the hypotenuse AC (c/2). Applying the Pythagorean Theorem in this smaller triangle:
OB^2 = OC^2 + BC^2
OB^2 = (c/2)^2 + (c/2)^2
OB^2 = c^2/4 + c^2/4
OB^2 = c^2/2
Taking the square root of both sides:
OB = c/√2
Now, we know that OA = OC = c/2 and OB = c/√2. Since c/√2 = (c/2) * √2, we can see that OB = OA = OC, and we have successfully shown that OA = OB = OC in the given triangle.
Answer:To show that OA = OB = OC in a right-angled triangle ABC with right angle at B, where O is the midpoint of the hypotenuse AC, you can use the Pythagorean Theorem.
Let's denote the length of the hypotenuse AC as c. Then, OA = OC = c/2, because O is the midpoint of AC.
Now, to show that OB = c/2, we can use the Pythagorean Theorem. In triangle ABC, we have:
AB^2 + BC^2 = AC^2
Since it's a right-angled triangle, AB^2 + BC^2 = c^2.
Since O is the midpoint of AC, we can say that AO = CO = c/2.
Now, we have a right-angled triangle OBC where OB is the hypotenuse, and BC is half of the hypotenuse AC (c/2). Applying the Pythagorean Theorem in this smaller triangle:
OB^2 = OC^2 + BC^2
OB^2 = (c/2)^2 + (c/2)^2
OB^2 = c^2/4 + c^2/4
OB^2 = c^2/2
Taking the square root of both sides:
OB = c/√2
Now, we know that OA = OC = c/2 and OB = c/√2. Since c/√2 = (c/2) * √2, we can see that OB = OA = OC, and we have successfully shown that OA = OB = OC in the given triangle.
Step-by-step explanation: