Construct any right-angle triangle ABC. Let angle A be the right angle. Construct a line AD
perpendicular to side BC.

To construct a right-angle triangle \( \triangle ABC \) with \( \angle A \) as the right angle and then construct a line \( AD \) perpendicular to side \( BC \), follow these steps:

### Step 1: Construct the right-angle triangle \( \triangle ABC \)
1. **Draw the base \( BC \):**
   - Use a straightedge to draw a line segment \( BC \). This will be the base of the triangle.
  
2. **Construct the right angle at \( A \):**
   - From point \( B \), use a compass to draw an arc above the line \( BC \).
   - Place the compass at point \( C \) and draw another arc that intersects the previous arc. Mark the intersection point as \( A \).
   - Draw line segments \( AB \) and \( AC \) to complete the right-angle triangle \( \triangle ABC \) where \( \angle A = 90^\circ \).

### Step 2: Construct the perpendicular line \( AD \)
1. **Find the midpoint of \( BC \) (optional for accuracy):**
   - Use a compass to find the midpoint \( M \) of segment \( BC \) by drawing arcs from \( B \) and \( C \) that intersect above and below \( BC \).
   - Connect these intersection points with a line segment. The point where this line crosses \( BC \) is the midpoint \( M \). \( M \) is not necessary for this construction but can help in checking perpendicularity.

2. **Construct the perpendicular \( AD \):**
   - From point \( A \), use a compass and straightedge to draw a perpendicular line to \( BC \). Extend this line until it intersects \( BC \). Label the intersection point as \( D \).

Now you have a right-angle triangle \( \triangle ABC \) with a line \( AD \) perpendicular to \( BC \).

Would you like further details on any part of the construction, or do you have any questions?

### 5 Relative Questions:
1. How do you construct the midpoint of a line segment using a compass?
2. What are the properties of the perpendicular bisector in a triangle?
3. How can you confirm that two lines are perpendicular in a geometric construction?
4. What are some methods to verify that \( \angle A \) is exactly \( 90^\circ \)?
5. How can the Pythagorean Theorem be applied to verify the sides of \( \triangle ABC \)?

### Tip:
When constructing perpendicular lines using a compass, always ensure your compass is set to a width greater than half the length of the segment to avoid errors in intersection points.

Learn how to construct a right-angle triangle ABC with angle A as the right angle and construct a perpendicular line AD to side BC. Follow step-by-step instructions and understand the geometric principles involved. Suitable for students in Grades 7-9 studying basic geometry constructions.

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