Geometry Name___________________ Altitude and Median ... - Free Printable

Problem Analysis:

The worksheet involves identifying and drawing specific geometric elements in triangles, such as angle bisectors, medians, perpendicular bisectors, and altitudes. Let's solve each part step by step.

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Part 1: Using the Diagram to Answer Questions 1–4

#### Question 1: Name an angle bisector in \( \Delta EFG \).
- An angle bisector is a line segment that divides an angle into two equal parts.
- In the given diagram, \( BD \) appears to be an angle bisector of \( \angle EBG \) because it splits the angle into two equal parts (indicated by the small arcs near \( \angle EBD \) and \( \angle DBG \)).

Answer: \( BD \)

#### Question 2: Name a median in \( \Delta EFG \).
- A median is a line segment joining a vertex of a triangle to the midpoint of the opposite side.
- In the diagram, \( EC \) appears to be a median because it connects vertex \( E \) to point \( C \), which is the midpoint of side \( FG \).

Answer: \( EC \)

#### Question 3: Name a perpendicular bisector in \( \Delta EFG \).
- A perpendicular bisector is a line that passes through the midpoint of a side and is perpendicular to that side.
- In the diagram, \( AD \) appears to be a perpendicular bisector of side \( EG \) because it intersects \( EG \) at its midpoint \( D \) and forms a right angle with \( EG \).

Answer: \( AD \)

#### Question 4: Name an altitude in \( \Delta EFG \).
- An altitude is a perpendicular segment from a vertex to the line containing the opposite side (or its extension).
- In the diagram, \( FB \) appears to be an altitude because it is perpendicular to side \( EG \) (indicated by the right angle symbol at \( B \)).

Answer: \( FB \)

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Part 2: Drawing All Altitudes in Each Triangle (Questions 5–7)

#### Question 5: Draw all altitudes in \( \Delta ABC \).
- Steps:
1. From vertex \( A \), draw a perpendicular to side \( BC \).
2. From vertex \( B \), draw a perpendicular to side \( AC \).
3. From vertex \( C \), draw a perpendicular to side \( AB \).

- Explanation: The altitudes will intersect at a single point (the orthocenter) for this acute triangle.

#### Question 6: Draw all altitudes in \( \Delta DEF \).
- Steps:
1. From vertex \( D \), draw a perpendicular to side \( EF \) (already given as a right angle).
2. From vertex \( E \), draw a perpendicular to side \( DF \).
3. From vertex \( F \), draw a perpendicular to side \( DE \).

- Explanation: Since \( \Delta DEF \) is a right triangle, one altitude is already given (the leg \( DE \)). The other two altitudes will intersect at the right angle vertex \( E \).

#### Question 7: Draw all altitudes in \( \Delta GHI \).
- Steps:
1. From vertex \( G \), draw a perpendicular to side \( HI \).
2. From vertex \( H \), draw a perpendicular to side \( GI \).
3. From vertex \( I \), draw a perpendicular to side \( GH \).

- Explanation: The altitudes will intersect at a single point (the orthocenter) for this obtuse triangle. One altitude may extend outside the triangle.

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Part 3: Drawing All Medians in Each Triangle (Questions 8–10)

#### Question 8: Draw all medians in \( \Delta ABC \).
- Steps:
1. Find the midpoint of side \( BC \) and draw a line from vertex \( A \) to this midpoint.
2. Find the midpoint of side \( AC \) and draw a line from vertex \( B \) to this midpoint.
3. Find the midpoint of side \( AB \) and draw a line from vertex \( C \) to this midpoint.

- Explanation: The medians will intersect at a single point (the centroid) for this triangle.

#### Question 9: Draw all medians in \( \Delta DEF \).
- Steps:
1. Find the midpoint of side \( EF \) and draw a line from vertex \( D \) to this midpoint.
2. Find the midpoint of side \( DF \) and draw a line from vertex \( E \) to this midpoint.
3. Find the midpoint of side \( DE \) and draw a line from vertex \( F \) to this midpoint.

- Explanation: The medians will intersect at a single point (the centroid) for this right triangle.

#### Question 10: Draw all medians in \( \Delta GHI \).
- Steps:
1. Find the midpoint of side \( HI \) and draw a line from vertex \( G \) to this midpoint.
2. Find the midpoint of side \( GI \) and draw a line from vertex \( H \) to this midpoint.
3. Find the midpoint of side \( GH \) and draw a line from vertex \( I \) to this midpoint.

- Explanation: The medians will intersect at a single point (the centroid) for this obtuse triangle.

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Final Answers:

1. Angle bisector: \( \boxed{BD} \)
2. Median: \( \boxed{EC} \)
3. Perpendicular bisector: \( \boxed{AD} \)
4. Altitude: \( \boxed{FB} \)

For questions 5–10, the solutions involve drawing the respective geometric elements as described above. If you need further clarification or detailed drawings, feel free to ask!

Final Boxed Answer for Part 1:
\[
\boxed{BD, EC, AD, FB}
\]