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WJEC MATHEMATICS INTERMEDIATE ANGLES AND GEOMETRY CONSTRUCTIONS AND LOCI 1 Contents Perpendicular Bisector of a Line Angle Bisector Drawing an angle of 60o Drawing an angle of 30o Constructing a Triangle - 3 Lengths Constructing a Triangle - 2 Sides, 1 Angle Constructing a Triangle - 1 Side, 2 Angles Locus Locus Example Credits WJEC Question bank http://www.wjec.co.uk/question-bank/question-search.html 2 Perpendicular Bisector of a Line Bisecting a line means to divide it evenly in two. Example Begin with a line, A B Step 1 Place your compass at point A, open it over halfway and draw an arc A B Step 2 Without altering the compass, place the compass at point B and draw another arc A B This has created an angle of 90o Step 3 Draw a line through both points of intersection. 3 Angle Bisector Bisecting an angle means dividing (splitting) the angle evenly in two Example Begin with an angle B A C Step 1 Place your compass at point A and draw an arc that crosses both lines B A C Step 2 Without changing the compass, place the compass at the two points of intersection and create two arcs that overlap B A C Step 3 Draw a line that originates at point A and goes through the two arcs that you have just drawn 4 Drawing an angle of 60o Begin with a line A B A B A B Step 1 Place your compass point on A and open the compass to the same length as the line AB. Draw an arc. Step 2 Without altering the compass, place the point on B and draw another arc. Step 3 Connect either point A or B to the point where the two arcs meet. This will give you an angle of 60o 60o A B If you connect point B to the intersection you will have an equilateral triangle. 5 Constructing an angle of 30o To construct an angle of 30o, construct a 60o angle and then bisect that angle as seen previously in this booklet. Practice drawing angles of 60o and 30o Exercise G19 1. Bisect the following lines 2. Bisect the following angles 6 Constructing a Triangle - Knowing All Lengths Construct an accurate drawing of a triangle with side lengths 10cm, 6cm, and 5cm. Step 1 Use a ruler to draw a line the length of a side given in the question. (10cm) 10cm Step 2 Open the compass to the length of the second side (6cm) and draw an arc from one end point 10cm Step 3 Open the compass to the length of the third side (5cm) and draw an arc from the other end point. 5cm 6cm 10cm Step 4 Join the end points to the intersection to create the triangle 7 Constructing a Triangle - Knowing 2 Sides, 1 Angle Construct a triangle with a length of 4.5cm, a length of 6.6cm, and an angle of 47 degrees. Step 1 Start by drawing a line equal to a length given in the question. 6.6cm Step 2 Place your protractor on one end of the line and make a mark at 47o x 6.6cm Step 3 Draw a line, the length given in the question (4.5cm) through the mark made in step 2 4.5cm x 47o 6.6cm Step 4 Join the two lines to make a triangle 8 Constructing a Triangle - Knowing 2 angles, 1 side Construct an accurate diagram of a triangle with a side length of 5cm, an angle of 45o, and an angle of 58o Step 1 Begin by drawing a line the length given in the question (5cm) 5cm Step 2 Place your protractor on one end and measure an angle given in the question (45o). Mark this with a cross. x 5cm Step 3 Place your protractor on the other side and measure the other angle given in the question (58o). Mark this angle with a cross. x x Step 4 5cm Draw a line from either end through the points made x x 5cm 9 Exercise G20 1. Construct a triangle with side lengths 12cm, 7cm, and 8cm 2. Construct a triangle with side lengths 15cm, 9cm, and 6cm 3. Construct a triangle with side lengths 10cm, 9cm, and 5cm 4. Construct a triangle with a side length 12cm, a side length 3cm, and an angle of 65o 5. Construct a triangle with a side length 8cm, a side length 6cm, and an angle of 73o 6. Construct a triangle with a side length 15cm, a side length 5cm, and an angle of 44o 7. Construct a triangle with a side length 10.5cm, an angle 49o, and an angle of 86o 8. Construct a triangle with a side length 8.6cm, an angle 57o, and an angle of 71o 9. Construct a triangle with a side length 4.3cm, an angle 33o, and an angle of 36o 10 Locus A locus is a path. A locus around a point is the path that remains a constant distance from the line at all times. Example 1 The locus of all points 3cm from A The locus of A all points 3cm less than 3cm from A Example 2 Shade the region that is closer to point A than point B A x x B Imagine AB is a line, we know how to bisect this A x x B This shaded region contains all points closer to A than B 11 Example 3 Shade the region that is closer to the line AB than the line AC B C A Begin by bisecting the angle B C A Now, shade all points closer to AB A B C 12 Example 4 Sketch the line that remains 5cm from the line AB Use a compass on either end and a ruler in the middle Example 5 Shade the region of points that satisfy the following conditions; Closer to AC than AB Less than 4cm from B Bisect the angle at A to see which side is Using a compass to closer to AC than AB draw a line 4cm from B Region that satisfies both conditions 13 Exam Questions G29 1. 2. 14 3. 4. 15 5. 16