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.

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 is perpendicular to the opposite side only in specific cases, such as in an isosceles triangle where the angle bisector of the vertex angle is also the altitude and median.

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, making it equidistant from both endpoints.

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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.

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.

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: Since \( K \) is the midpoint and \( \overline{RK} \) is perpendicular to \( \overline{ST} \), \( \overline{RK} \) is 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: If \( \overline{RK} \) is both a median and an altitude, then \( K \) is the midpoint of \( \overline{ST} \) and \( \overline{RK} \perp \overline{ST} \). This means \( SK = KT \) and \( \angle RKS = \angle RKT = 90^\circ \). Also, \( RK \) is common to both triangles. Therefore, \( \Delta RSK \cong \Delta RTK \) by SAS.
- b. \( \Delta RST \) is a(n) ___________ triangle.
- Answer: Isosceles
- Explanation: If \( \overline{RK} \) is both a median and an altitude, then \( \Delta RST \) must be 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 line 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 \( \angle ACB \).

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

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, altitude, and perpendicular bisector of the base \( \overline{DE} \). Therefore, \( \overline{FX} \) is the median.

15. Would the altitude drawn from \( F \) to \( \overline{DE} \) be the same segment \( \overline{FX} \)?
- Answer: Yes
- Explanation: Similar to the previous explanation, in an isosceles triangle with \( DF = EF \), the angle bisector \( \overline{FX} \) is also the altitude from \( F \) to \( \overline{DE} \).

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Final Answers:

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