Circumcenter of a right triangle | Geometry | Khan Academy
Khan Academy・5 minutes read
The circumcenter of a right triangle is proven to be the midpoint of the hypotenuse by showing that the distances from the midpoint to the endpoints are equal, while the ratio of corresponding sides in similar triangles is constant, confirming the circumcenter as equidistant from all vertices. This establishes the relationship between the circumcenter and the midpoint of the hypotenuse in a right triangle.
Insights
- The circumcenter of a right triangle is located at the midpoint of the hypotenuse, as it is equidistant from all vertices due to the construction of perpendicular bisectors, demonstrating the unique geometric relationship between these elements.
- Similarity between triangles OBM and ABC reveals the constancy of ratios between corresponding sides in similar triangles, enabling the calculation of distances and ultimately confirming the circumcenter's position as equidistant from all vertices, solidifying its role within the right triangle's geometry.
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Recent questions
How is the circumcenter of a right triangle determined?
By constructing perpendicular bisector of a leg.
What is the significance of similar triangles in geometry?
Ratios of corresponding sides are constant.
How is the circumcenter related to the vertices of a triangle?
Circumcenter is equidistant from all vertices.
What confirms the midpoint of the hypotenuse as the circumcenter?
Equidistance from all vertices.
How does constructing perpendicular bisectors aid in geometry?
Determines key points like circumcenter.
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Summary
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Circumcenter and Midpoint Relationship in Triangles
- The circumcenter of a right triangle is proven to be the midpoint of the hypotenuse by constructing the perpendicular bisector of one of the legs, showing that the distances from the midpoint to the endpoints are equal.
- Triangle OBM is shown to be similar to triangle ABC, leading to the conclusion that the ratio of corresponding sides in similar triangles is constant.
- By calculating the ratios of sides in the similar triangles, it is determined that the circumcenter is equidistant from all vertices, establishing it as the circumcenter of the right triangle.
- The midpoint of the hypotenuse is confirmed as the circumcenter, as it is equidistant from all vertices of the triangle, proving the relationship between the circumcenter and the midpoint of the hypotenuse in a right triangle.
