Understanding the Circumcenter of Triangle DEF | Statements that Must Be True

Point H is the circumcenter of triangle DEF.Which must be true? Check all that apply.

To determine which statements must be true about the circumcenter of triangle DEF, we need to understand what a circumcenter is and how it relates to a triangle

To determine which statements must be true about the circumcenter of triangle DEF, we need to understand what a circumcenter is and how it relates to a triangle.

Definition: The circumcenter of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle. It is equidistant from the three vertices of the triangle.

Now, let’s analyze the statements:

1. Point H is equidistant from the three vertices of triangle DEF.
This statement is true by definition. The circumcenter is equidistant from all three vertices of the triangle.

2. The distance from point H to any vertex of triangle DEF is the same as the distance from point H to the midpoint of the opposite side.
This statement is also true. The perpendicular bisectors of the sides of the triangle intersect at the circumcenter, making the distance from H to any vertex equal to the distance from H to the midpoint of the opposite side.

3. Point H lies within the triangle DEF.
This statement may or may not be true. The circumcenter can lie inside, on the boundary, or outside the triangle, depending on the type of triangle. For example, in an acute triangle, the circumcenter will be inside the triangle. In an obtuse triangle, the circumcenter will be outside the triangle. And in a right triangle, the circumcenter will be on the midpoint of the hypotenuse.

4. The circumradius is the distance from point H to any vertex of triangle DEF.
This statement is true. The distance from the circumcenter to any vertex of the triangle is known as the circumradius. Since H is the circumcenter, it follows that the circumradius is the distance from H to any vertex.

So, the statements that must be true about the circumcenter of triangle DEF are:
– Point H is equidistant from the three vertices of triangle DEF.
– The distance from point H to any vertex of triangle DEF is the same as the distance from point H to the midpoint of the opposite side.
– The circumradius is the distance from point H to any vertex of triangle DEF.

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