Question
Point H is the circumcenter of triangle DEF.
Point H is the circumcenter of triangle D E F. Lines are drawn from each point of the triangle to point H. Lines are drawn from point H to each side of the triangle to form right angles and line segments H M, H N, and H L.
Which must be true? Select two options.
Point H is the center of the circle that passes through points D, E, and F.
Point H is the center of the circle that passes through points L, M, and N.
Line segment H E is-congruent-to line segment H D
Line segment L H is-congruent-to line segment N H
Line segment F L is-congruent-to line segment F N
Point H is the circumcenter of triangle D E F. Lines are drawn from each point of the triangle to point H. Lines are drawn from point H to each side of the triangle to form right angles and line segments H M, H N, and H L.
Which must be true? Select two options.
Point H is the center of the circle that passes through points D, E, and F.
Point H is the center of the circle that passes through points L, M, and N.
Line segment H E is-congruent-to line segment H D
Line segment L H is-congruent-to line segment N H
Line segment F L is-congruent-to line segment F N
Answers
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Answered by
GPT-4o mini
Given that point H is the circumcenter of triangle DEF, we know the following properties about the circumcenter and its relationship to the triangle and the circle that passes through its vertices:
1. **Point H is the center of the circle that passes through points D, E, and F.**
- This statement is true because by definition, the circumcenter of a triangle is the point that is equidistant from all three vertices, making it the center of the circumcircle (the circle that passes through the vertices of the triangle).
2. **Point H is the center of the circle that passes through points L, M, and N.**
- This statement is **not necessarily true**. Points L, M, and N are the feet of the perpendiculars dropped from H to the sides of triangle DEF, and there is no given information that suggests they are equidistant from H.
3. **Line segment HE is congruent to line segment HD.**
- This statement is **true** because both segments are radii of the circumcircle of triangle DEF. Since H is equidistant from the vertices D, E, and F, HE = HD.
4. **Line segment LH is congruent to line segment NH.**
- This statement is **not necessarily true**. L and N are the feet of the perpendiculars from H to the sides of the triangle, and we cannot infer that they are equidistant from H.
5. **Line segment FL is congruent to line segment FN.**
- This statement is also **not necessarily true**. Similar to the previous options, L and N are specific points on different lines, and there is no property guaranteeing their distances to F are equal.
Therefore, the two statements that must be true are:
- **Point H is the center of the circle that passes through points D, E, and F.**
- **Line segment HE is congruent to line segment HD.**
1. **Point H is the center of the circle that passes through points D, E, and F.**
- This statement is true because by definition, the circumcenter of a triangle is the point that is equidistant from all three vertices, making it the center of the circumcircle (the circle that passes through the vertices of the triangle).
2. **Point H is the center of the circle that passes through points L, M, and N.**
- This statement is **not necessarily true**. Points L, M, and N are the feet of the perpendiculars dropped from H to the sides of triangle DEF, and there is no given information that suggests they are equidistant from H.
3. **Line segment HE is congruent to line segment HD.**
- This statement is **true** because both segments are radii of the circumcircle of triangle DEF. Since H is equidistant from the vertices D, E, and F, HE = HD.
4. **Line segment LH is congruent to line segment NH.**
- This statement is **not necessarily true**. L and N are the feet of the perpendiculars from H to the sides of the triangle, and we cannot infer that they are equidistant from H.
5. **Line segment FL is congruent to line segment FN.**
- This statement is also **not necessarily true**. Similar to the previous options, L and N are specific points on different lines, and there is no property guaranteeing their distances to F are equal.
Therefore, the two statements that must be true are:
- **Point H is the center of the circle that passes through points D, E, and F.**
- **Line segment HE is congruent to line segment HD.**
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