Geometric constructions
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This document provides worked examples for constructing geometric shapes using a ruler, pencil, and compass. It includes three examples of constructing different shapes step-by-step: 1) a triangle with given side lengths and angles, 2) a triangle with a perpendicular line segment, and 3) a trapezoid with given side lengths and angles. The examples show the step-by-step construction of each shape and measure any required lengths or angles.
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2) If two sides of a triangle are congruent, then the angles opposite those sides are also congruent. Similarly, if two angles are congruent, the sides opposite them are also congruent.
3) In a triangle, if one side is longer than the other, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.
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The scale of 1:250,000 as a statement is:
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Plastica 1º eso
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Geometric constructions
- 2. EXAMPLES
- 3. EXAMPLE 1 • Using a ruler, a pencil and a pair of compasses, construct triangle 𝐸𝐹𝐺, with 𝐸𝐺 = 6𝑐𝑚, < 𝐹𝐸𝐺 = 60° 𝑎𝑛𝑑 < 𝐸𝐺𝐹 = 90°. • Measure and state the length of 𝐸𝐹. • Measure and state the size of < 𝐸𝐹𝐺.
- 4. EXAMPLE 1- SOLUTION • Using a ruler, a pencil and a pair of compasses, construct triangle 𝐸𝐹𝐺, with 𝐸𝐺 = 6𝑐𝑚, < 𝐹𝐸𝐺 = 60° 𝑎𝑛𝑑 < 𝐸𝐺𝐹 = 90°. Step 1: Draw the line segment 𝐸𝐺 = 6𝑐𝑚. Step 2: Construct < 𝐹𝐸𝐺 = 60°
- 5. EXAMPLE 1- SOLUTION CONT’D • Using a ruler, a pencil and a pair of compasses, construct triangle 𝐸𝐹𝐺, with 𝐸𝐺 = 6𝑐𝑚, < 𝐹𝐸𝐺 = 60° 𝑎𝑛𝑑 < 𝐸𝐺𝐹 = 90°. Step 3: Construct < 𝐸𝐺𝐹 = 90°. Step 4: Join the lines to complete the triangle.
- 6. EXAMPLE 1- SOLUTION CONT’D Measure and state the length of 𝐸𝐹. Measure and state the size of < 𝐸𝐹𝐺. 𝑬𝑭 = 𝟏𝟐 𝒄𝒎 < 𝑬𝑭𝑮 = 𝟑𝟎°
- 7. EXAMPLE 2 Using a ruler, a pencil and a pair of compasses, construct • a triangle 𝑃𝑄𝑅, with 𝑃𝑄 = 8𝑐𝑚, 𝑃𝑅 = 6 𝑐𝑚 𝑎𝑛𝑑 𝑎𝑛𝑔𝑙𝑒 𝑃 = 60°. • a line segment 𝑅𝑋, which is perpendicular to 𝑃𝑄 and meets 𝑃𝑄 at 𝑋. • Measure and state the size of < 𝑄𝑅𝑋.
- 8. EXAMPLE 2- SOLUTION Using a ruler, a pencil and a pair of compasses, construct • a triangle 𝑃𝑄𝑅, with 𝑃𝑄 = 8𝑐𝑚, 𝑃𝑅 = 6 𝑐𝑚 𝑎𝑛𝑑 𝑎𝑛𝑔𝑙𝑒 𝑃 = 60°. Step 1: Draw the line segment 𝑃𝑄 = 8𝑐𝑚. Step 2: Construct 𝑎𝑛𝑔𝑙𝑒 𝑃 = 60°
- 9. EXAMPLE 2- SOLUTION CONT’D. Using a ruler, a pencil and a pair of compasses, construct • a triangle 𝑃𝑄𝑅, with 𝑃𝑄 = 8𝑐𝑚, 𝑃𝑅 = 6 𝑐𝑚 𝑎𝑛𝑑 𝑎𝑛𝑔𝑙𝑒 𝑃 = 60°. Step 3: Draw the line segment 𝑃𝑅 = 6𝑐𝑚. Step 4: Join 𝑄 to 𝑅 to complete the triangle.
- 10. EXAMPLE 2- SOLUTION CONT’D. Using a ruler, a pencil and a pair of compasses, construct • a line segment 𝑅𝑋, which is perpendicular Measure and state the size of < 𝑄𝑅𝑋. to 𝑃𝑄 and meets 𝑃𝑄 at 𝑋. < 𝑸𝑹𝑿 = 𝟒𝟒°
- 11. EXAMPLE 3 Using a ruler, a pencil and a pair of compasses, construct • The trapezium 𝐴𝐵𝐶𝐷, with 𝐴𝐵 = 8𝑐𝑚, 𝐵𝐴𝐷 = 60°, 𝐴𝐷 = 6 𝑐𝑚, 𝐴𝐵𝐶 = 90°, and 𝐴𝐵 parallel to 𝐶𝐷.
- 12. EXAMPLE 3- SOLUTION Using a ruler, a pencil and a pair of compasses, construct • The trapezium 𝐴𝐵𝐶𝐷, with 𝐴𝐵 = 8𝑐𝑚, 𝐵𝐴𝐷 = 60°, 𝐴𝐷 = 6 𝑐𝑚, 𝐴𝐵𝐶 = 90°, and 𝐴𝐵 parallel to 𝐶𝐷. Step 1: Draw the line segment 𝐴𝐵 = 8𝑐𝑚. Step 2: Construct 𝐵𝐴𝐷 = 60°
- 13. EXAMPLE 3- SOLUTION CONT’D. Using a ruler, a pencil and a pair of compasses, construct • The trapezium 𝐴𝐵𝐶𝐷, with 𝐴𝐵 = 8𝑐𝑚, 𝐵𝐴𝐷 = 60°, 𝐴𝐷 = 6 𝑐𝑚, 𝐴𝐵𝐶 = 90°, and 𝐴𝐵 parallel to 𝐶𝐷. Step 3: Construct 𝐴𝐵𝐶 = 90° Step 4: Mark off 6 𝑐𝑚 along the ray from point A, to label point 𝐷.
- 14. EXAMPLE 3- SOLUTION CONT’D. Using a ruler, a pencil and a pair of compasses, construct • The trapezium 𝐴𝐵𝐶𝐷, with 𝐴𝐵 = 8𝑐𝑚, 𝐵𝐴𝐷 = 60°, 𝐴𝐷 = 6 𝑐𝑚, 𝐴𝐵𝐶 = 90°, and 𝐴𝐵 parallel to 𝐶𝐷. Step 5: Draw line segment 𝐶𝐷, parallel to 𝐴𝐵 to complete the trapezium.
- 15. THE END.