contruct a 45 angle by first constructing a perpendicular bisector of a segment and then bisecting one of the angles

To construct a 45° angle by first constructing a perpendicular bisector of a segment and then bisecting one of the angles, follow these steps:

### Steps:

1. **Draw a Segment:**
   Draw a line segment \( AB \) of any length using a ruler.

2. **Construct the Perpendicular Bisector:**
   - Place the compass point on point \( A \) and draw an arc above and below the line segment \( AB \) with a radius greater than half the length of the segment.
   - Without changing the radius, place the compass point on point \( B \) and draw another arc above and below \( AB \), intersecting the previous arcs.
   - Label the points of intersection of the arcs above and below \( AB \) as \( P \) and \( Q \).
   - Draw a straight line through points \( P \) and \( Q \). This is the perpendicular bisector of \( AB \), which forms a 90° angle with \( AB \) at its midpoint \( M \).

3. **Label the Intersection Point:**
   The perpendicular bisector intersects segment \( AB \) at its midpoint \( M \), creating two equal halves of the segment and forming a 90° angle.

4. **Bisect the 90° Angle:**
   - To bisect the 90° angle formed between the perpendicular bisector and segment \( AB \), place the compass point at point \( M \) and draw an arc that intersects both the perpendicular bisector and \( AB \).
   - Label the points of intersection as \( C \) (on the perpendicular bisector) and \( D \) (on segment \( AB \)).
   - Now, place the compass point on \( C \) and draw an arc inside the 90° angle. Without changing the compass width, place the compass point on \( D \) and draw another arc inside the 90° angle, intersecting the previous arc. Label this intersection point as \( E \).
   - Draw a straight line from point \( M \) through point \( E \). This line bisects the 90° angle, forming two 45° angles.

Now, the angle formed between \( AB \) and the bisector line \( ME \) is 45°.

### Summary:
- Step 1: Draw a segment.
- Step 2: Construct its perpendicular bisector.
- Step 3: Bisect the 90° angle to get a 45° angle.

Would you like more details on any of these steps?

---

### Related Questions:
1. How can you verify that the constructed angle is exactly 45°?
2. How do you construct a 60° angle using a compass?
3. How can you construct a perpendicular line to a given line without using a protractor?
4. What is the geometrical significance of bisecting angles in triangle constructions?
5. How would you construct a 30° angle by bisecting a 60° angle?

**Tip:** Always make sure your compass is firm and doesn’t shift when drawing arcs for accurate constructions.

Construct a 45° angle by first constructing a perpendicular bisector of a segment and then bisecting one of the angles.

Learn how to construct a 45° angle by first creating a perpendicular bisector of a segment and then bisecting one of the angles. This step-by-step guide uses basic geometric principles and theorems to ensure accuracy. Ideal for middle school students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation