Triangle Constructions

Overview

Constructing triangles involves creating shapes that meet specific conditions regarding sides and angles. These constructions can include different combinations of given elements, such as three sides, two sides and an angle, or information about right-angled triangles. A compass and straight edge are used for all constructions.

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Triangle Given Three Sides (SSS Construction)

Objective: To construct a triangle when the lengths of all three sides are known.

Method

1. Draw the base of the triangle with the given length.
2. Using the compass, set the radius to the length of the second side. Place the compass at one endpoint of the base and draw an arc.
3. Repeat with the third side from the other endpoint of the base.
4. The intersection of the arcs marks the third vertex of the triangle.
5. Connect this vertex to the endpoints of the base. Result: A triangle with the given side lengths is constructed.

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Triangle Given Two Sides and the Included Angle (SAS Construction)

Objective: To construct a triangle when two sides and the included angle are known.

Method

1. Draw the base of the triangle with the length of the first side.
2. At one endpoint of the base, use a protractor to construct the given included angle.
3. Set the compass to the length of the second side. Place the compass at the vertex of the angle and draw an arc that intersects the second arm of the angle.
4. Connect the intersection point to the endpoints of the base. Result: A triangle with the given sides and included angle is constructed.

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Triangle Given Two Angles and the Included Side (ASA Construction)

Objective: To construct a triangle when two angles and the included side are known.

Method

1. Draw the given side as the base of the triangle.
2. At each endpoint of the base, use a protractor to construct the given angles.
3. Extend the arms of the angles until they meet at a point, forming the third vertex of the triangle. Result: A triangle with the given angles and side is constructed.

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Right-Angled Triangle Given the Hypotenuse and One Other Side

Objective: To construct a right triangle when the hypotenuse and one side are known.

Method

1. Draw the given hypotenuse as the base.
2. Use the compass to set the length of the given side. Place the compass at one endpoint of the base and draw an arc.
3. At the same endpoint, construct a perpendicular line using a compass and straight edge.
4. The arc and the perpendicular line intersect at the third vertex of the triangle.
5. Connect this vertex to the endpoints of the hypotenuse. Result: A right triangle with the given hypotenuse and side is constructed.

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Right-Angled Triangle Given One Side and One Acute Angle

Objective: To construct a right triangle when one side and an acute angle are known.

Method

1. Draw the given side as the base of the triangle.
2. At one endpoint of the base, use a protractor to construct the given acute angle.
3. Extend the arm of the angle to form the hypotenuse.
4. At the other endpoint of the base, draw a perpendicular line using a compass and straight edge. The perpendicular line intersects the extended arm, forming the third vertex.
5. Connect this vertex to the endpoints of the base. Result: A right triangle with the given side and acute angle is constructed.

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Summary

1. SSS Construction: Triangle with three known side lengths.
2. SAS Construction: Triangle with two sides and the included angle.
3. ASA Construction: Triangle with two angles and the included side.
4. Right-Angled Triangle (Hypotenuse and Side): Constructs a right triangle with given hypotenuse and one side.
5. Right-Angled Triangle (Side and Angle): Constructs a right triangle with a given side and acute angle. These constructions are essential for solving problems in geometry and for creating accurate geometric diagrams.