6-1 Secondary Parts of A Triangle | PDF | Perpendicular ... - Free Printable

Problem Analysis and Solution

The worksheet focuses on understanding medians, altitudes, and angle bisectors in triangles. Let's solve each part step by step.

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#### Section 1: Always, Sometimes, or Never

1. An altitude is ___________ perpendicular to the opposite side.
- Answer: Always
- Explanation: By definition, an altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side (or its extension).

2. A median is ___________ perpendicular to the opposite side.
- Answer: Sometimes
- Explanation: A median connects a vertex to the midpoint of the opposite side. It is not always perpendicular to the opposite side unless the triangle is isosceles or equilateral, where the median coincides with the altitude.

3. An altitude is ___________ an angle bisector.
- Answer: Sometimes
- Explanation: An altitude is an angle bisector only in special cases, such as in an isosceles triangle where the altitude from the vertex angle also bisects the base.

4. An angle bisector is ___________ perpendicular to the opposite side.
- Answer: Sometimes
- Explanation: An angle bisector divides the angle into two equal parts but is not always perpendicular to the opposite side. It is perpendicular only in specific cases, such as in an isosceles triangle where the angle bisector of the vertex angle is also the altitude.

5. A perpendicular bisector of a segment is ___________ equidistant from the endpoints of the segment.
- Answer: Always
- Explanation: By definition, a perpendicular bisector of a segment is a line that is perpendicular to the segment and passes through its midpoint. Every point on this line is equidistant from the endpoints of the segment.

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#### Section 2: Complete using the diagram to the right

The diagram shows a triangle \( \Delta RST \) with point \( K \) on side \( \overline{ST} \).

6. If \( K \) is the midpoint of \( \overline{ST} \), then \( \overline{RK} \) is called a(n) ___________ of \( \Delta RST \).
- Answer: Median
- Explanation: A median connects a vertex to the midpoint of the opposite side. Since \( K \) is the midpoint of \( \overline{ST} \), \( \overline{RK} \) is a median.

7. If \( \overline{RK} \perp \overline{ST} \), then \( \overline{RK} \) is called a(n) ___________ of \( \Delta RST \).
- Answer: Altitude
- Explanation: An altitude is a line segment from a vertex perpendicular to the opposite side. If \( \overline{RK} \perp \overline{ST} \), then \( \overline{RK} \) is an altitude.

8. If \( K \) is the midpoint of \( \overline{ST} \) and \( \overline{RK} \perp \overline{ST} \), then \( \overline{RK} \) is called the ___________ of \( \overline{ST} \).
- Answer: Perpendicular Bisector
- Explanation: If \( K \) is the midpoint of \( \overline{ST} \) and \( \overline{RK} \perp \overline{ST} \), then \( \overline{RK} \) is both a median and an altitude, making it the perpendicular bisector of \( \overline{ST} \).

9. If \( \overline{RK} \) is both an altitude and a median of \( \Delta RST \), then:
- a. \( \Delta RSK \cong \Delta RTK \) by ___________
- Answer: SAS (Side-Angle-Side)
- Explanation: Since \( \overline{RK} \) is both a median and an altitude:
- \( SK = KT \) (median property)
- \( \angle RKS = \angle RKT = 90^\circ \) (altitude property)
- \( RK = RK \) (common side)
- Therefore, \( \Delta RSK \cong \Delta RTK \) by SAS congruence.
- b. \( \Delta RST \) is a(n) ___________ triangle.
- Answer: Isosceles
- Explanation: If \( \overline{RK} \) is both a median and an altitude, then \( \Delta RST \) is isosceles with \( RS = RT \).

10. If \( R \) is on the perpendicular bisector of \( \overline{ST} \), then \( R \) is equidistant from ________ and ________. Thus, ________ = ________.
- Answer: \( S \) and \( T \); \( RS = RT \)
- Explanation: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Therefore, if \( R \) is on the perpendicular bisector of \( \overline{ST} \), then \( RS = RT \).

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#### Section 3: Refer to \( \Delta ABC \) and name the following

The diagram shows \( \Delta ABC \) with various lines drawn.

11. a median of \( \Delta ABC \)
- Answer: \( \overline{CX} \)
- Explanation: A median connects a vertex to the midpoint of the opposite side. Here, \( X \) is the midpoint of \( \overline{AB} \), so \( \overline{CX} \) is a median.

12. an altitude of \( \Delta ABC \)
- Answer: \( \overline{CD} \)
- Explanation: An altitude is a perpendicular segment from a vertex to the opposite side (or its extension). Here, \( \overline{CD} \perp \overline{AB} \), so \( \overline{CD} \) is an altitude.

13. a bisector of an angle of \( \Delta ABC \)
- Answer: \( \overline{CE} \)
- Explanation: An angle bisector divides an angle into two equal parts. Here, \( \overline{CE} \) bisects \( \angle ACB \), so \( \overline{CE} \) is an angle bisector.

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#### Section 4: Given \( \Delta DEF \) is isosceles with \( DF = EF \); \( \overline{FX} \) bisects \( \angle DFE \)

The diagram shows \( \Delta DEF \) with \( DF = EF \) and \( \overline{FX} \) bisecting \( \angle DFE \).

14. Would the median drawn from \( F \) to \( \overline{DE} \) be the same segment \( \overline{FX} \)?
- Answer: Yes
- Explanation: In an isosceles triangle where \( DF = EF \), the angle bisector of the vertex angle (\( \angle DFE \)) is also the median to the base (\( \overline{DE} \)). Since \( \overline{FX} \) bisects \( \angle DFE \), it is also the median.

15. Would the altitude drawn from \( F \) to \( \overline{DE} \) be the same segment \( \overline{FX} \)?
- Answer: Yes
- Explanation: In an isosceles triangle where \( DF = EF \), the angle bisector of the vertex angle (\( \angle DFE \)) is also the altitude to the base (\( \overline{DE} \)). Since \( \overline{FX} \) bisects \( \angle DFE \), it is also the altitude.

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Final Answers:

\[
\boxed{
\begin{array}{l}
1. \text{Always} \\
2. \text{Sometimes} \\
3. \text{Sometimes} \\
4. \text{Sometimes} \\
5. \text{Always} \\
6. \text{Median} \\
7. \text{Altitude} \\
8. \text{Perpendicular Bisector} \\
9. \text{a. SAS; b. Isosceles} \\
10. \text{S and T; RS = RT} \\
11. \text{\(\overline{CX}\)} \\
12. \text{\(\overline{CD}\)} \\
13. \text{\(\overline{CE}\)} \\
14. \text{Yes} \\
15. \text{Yes}
\end{array}
}
\]