can guide you through the proof for this statement. To show that OA = OB = OC, you can use the Pythagorean Theorem and properties of midpoints. Here's a step-by-step explanation:
Draw the right-angled triangle ABC with right angle at B.
Mark point O as the midpoint of hypotenuse AC. This means AO = OC because O is the midpoint.
Now, let's prove that OA = OB:
Draw line segment BO.
Since B is a right angle, you have a right triangle OBC.
By the Pythagorean Theorem, in right triangle OBC:
OB^2 + BC^2 = OC^2
Now, let's use the fact that OC = AO:
Replace OC with AO in the equation from step 3:
OB^2 + BC^2 = AO^2
However, by the Pythagorean Theorem in right triangle ABC:
AB^2 + BC^2 = AC^2
Since AB = AO (because A is also a midpoint of AC), you can replace AB with AO in the equation from step 5:
AO^2 + BC^2 = AC^2
Now, you have two equations:
AO^2 + BC^2 = AO^2 + BC^2 (from step 4)
AO^2 + BC^2 = AC^2 (from step 6)
Since both equations are equal, you can conclude that AO^2 + BC^2 = AO^2 + BC^2. Therefore, AO = OB.
Since AO = OB and AO = OC (because O is the midpoint of AC), you have shown that OA = OB = OC.
I hope this helps you understand and prove the statement. Feel free to write this out in your notebook for reference.
Answer: Can guide you through the proof for this statement. To show that OA = OB = OC, you can use the Pythagorean Theorem and properties of midpoints. Here's a step-by-step explanation:
Draw the right-angled triangle ABC with right angle at B.
Mark point O as the midpoint of hypotenuse AC. This means AO = OC because O is the midpoint.
Now, let's prove that OA = OB:
Draw line segment BO.
Since B is a right angle, you have a right triangle OBC.
By the Pythagorean Theorem, in right triangle OBC:
OB^2 + BC^2 = OC^2
Now, let's use the fact that OC = AO:
Replace OC with AO in the equation from step 3:
OB^2 + BC^2 = AO^2
However, by the Pythagorean Theorem in right triangle ABC:
AB^2 + BC^2 = AC^2
Since AB = AO (because A is also a midpoint of AC), you can replace AB with AO in the equation from step 5:
AO^2 + BC^2 = AC^2
Now, you have two equations:
AO^2 + BC^2 = AO^2 + BC^2 (from step 4)
AO^2 + BC^2 = AC^2 (from step 6)
Since both equations are equal, you can conclude that AO^2 + BC^2 = AO^2 + BC^2. Therefore, AO = OB.
Since AO = OB and AO = OC (because O is the midpoint of AC), you have shown that OA = OB = OC.
I hope this helps you understand and prove the statement. Feel free to write this out in your notebook for reference.
Answers & Comments
Answer:
can guide you through the proof for this statement. To show that OA = OB = OC, you can use the Pythagorean Theorem and properties of midpoints. Here's a step-by-step explanation:
Draw the right-angled triangle ABC with right angle at B.
Mark point O as the midpoint of hypotenuse AC. This means AO = OC because O is the midpoint.
Now, let's prove that OA = OB:
Draw line segment BO.
Since B is a right angle, you have a right triangle OBC.
By the Pythagorean Theorem, in right triangle OBC:
OB^2 + BC^2 = OC^2
Now, let's use the fact that OC = AO:
Replace OC with AO in the equation from step 3:
OB^2 + BC^2 = AO^2
However, by the Pythagorean Theorem in right triangle ABC:
AB^2 + BC^2 = AC^2
Since AB = AO (because A is also a midpoint of AC), you can replace AB with AO in the equation from step 5:
AO^2 + BC^2 = AC^2
Now, you have two equations:
AO^2 + BC^2 = AO^2 + BC^2 (from step 4)
AO^2 + BC^2 = AC^2 (from step 6)
Since both equations are equal, you can conclude that AO^2 + BC^2 = AO^2 + BC^2. Therefore, AO = OB.
Since AO = OB and AO = OC (because O is the midpoint of AC), you have shown that OA = OB = OC.
I hope this helps you understand and prove the statement. Feel free to write this out in your notebook for reference.
Answer: Can guide you through the proof for this statement. To show that OA = OB = OC, you can use the Pythagorean Theorem and properties of midpoints. Here's a step-by-step explanation:
Draw the right-angled triangle ABC with right angle at B.
Mark point O as the midpoint of hypotenuse AC. This means AO = OC because O is the midpoint.
Now, let's prove that OA = OB:
Draw line segment BO.
Since B is a right angle, you have a right triangle OBC.
By the Pythagorean Theorem, in right triangle OBC:
OB^2 + BC^2 = OC^2
Now, let's use the fact that OC = AO:
Replace OC with AO in the equation from step 3:
OB^2 + BC^2 = AO^2
However, by the Pythagorean Theorem in right triangle ABC:
AB^2 + BC^2 = AC^2
Since AB = AO (because A is also a midpoint of AC), you can replace AB with AO in the equation from step 5:
AO^2 + BC^2 = AC^2
Now, you have two equations:
AO^2 + BC^2 = AO^2 + BC^2 (from step 4)
AO^2 + BC^2 = AC^2 (from step 6)
Since both equations are equal, you can conclude that AO^2 + BC^2 = AO^2 + BC^2. Therefore, AO = OB.
Since AO = OB and AO = OC (because O is the midpoint of AC), you have shown that OA = OB = OC.
I hope this helps you understand and prove the statement. Feel free to write this out in your notebook for reference.
Step-by-step explanation: