Using a ruler and a pair of compasses only, construct an angle of 90° at a point on a straight line. Then, bisect the 90° angle to obtain an angle of 45°.
Answer (2)
Draw a straight line and mark a point A on it.
Construct a semicircle centered at A, intersecting the line at B and C.
Construct a perpendicular at A, creating a 90° angle DAC.
Bisect the 90° angle DAC to obtain a 45° angle FAC.
Explanation
Understanding the problem Let's start by understanding the problem. We need to construct a 45-degree angle using only a ruler and compass. This involves first constructing a 90-degree angle and then bisecting it.
Constructing a 90-degree angle
Draw a straight line: Use the ruler to draw a straight line on a piece of paper. Mark a point A on this line. This will be the vertex of our angle.
Construct a semicircle: Place the compass at point A and draw a semicircle that intersects the straight line at two points, B and C. Make sure B and C are equidistant from A.
Construct the perpendicular:
Open the compass to a radius greater than half the distance BC. Place the compass at point B and draw an arc above the line.
Without changing the compass radius, place the compass at point C and draw another arc that intersects the previous arc. Let's call the intersection point D.
Draw the 90° angle: Use the ruler to draw a straight line from point A to point D. The angle DAC is a 90° angle.
Bisecting the 90-degree angle
Bisect the 90° angle:
Place the compass at point B and draw an arc that intersects AD. Call this point E.
Place the compass at point E and draw an arc.
Place the compass at point C and draw an arc that intersects the previous arc. Call this intersection point F.
Drawing the 45-degree angle
Draw the 45° angle: Use the ruler to draw a straight line from point A to point F. The angle FAC is a 45° angle, as it bisects the 90° angle DAC.
Conclusion Therefore, by following these steps, you can construct a 45-degree angle at a point on a straight line using only a ruler and a pair of compasses.
Examples
In woodworking or construction, creating precise angles is crucial for accurate cuts and stable structures. Constructing a 45-degree angle is essential for creating miter joints in picture frames or furniture. By using a ruler and compass, you can accurately mark and cut the wood at a 45-degree angle, ensuring a perfect fit and a professional finish. This method provides a reliable way to create accurate angles without relying on protractors or other measuring tools.
You can construct a 90° angle by drawing a straight line and marking a point, then using a semicircle to find points for a perpendicular line. To bisect this angle into 45°, further arcs and intersections are used. This process demonstrates precise angle construction techniques using only basic tools.
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