Similarity, Congruence & Proof 1 Similarity, Congruence & Proof

Constructions Standard: MCC9-12.G.CO.12 make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc). Copying a segment; copying an angle, bisecting a segment; bisecting an angle, constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. MCC9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Essential Question: How do I construct geometric figures?

1 Bounded Book Constructions Standard: MCC9-12.G.CO.12 make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc). Copying a segment; copying an angle, bisecting a segment; bisecting an angle, constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. MCC9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Essential Question: How do I construct geometric figures? 1

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7 Bounded Book Construct a perpendicular bisector of a line segment  Place the compasses on one end of the line segment.  Set the compasses' width to a approximately two thirds the line length. The actual width does not matter. Without changing the compasses' width, draw an arc above and below the line.  Again without changing the compasses' width, place the compasses' point on the other end of the line. Draw an arc above and below the line so that the arcs cross the first two.  Using a straightedge, draw a line between the points where the arcs intersect. Done. This line is perpendicular to the first line and bisects it (cuts it at the exact midpoint of the line). You Try! Construct the perpendicular bisector of the line segment. 7

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9 Bounded Book Construct an equilateral triangle You Try! Starting with the line segment AB which is the length of the sides of the desired equilateral triangle, pick a point P that will be one vertex of the finished triangle. Place the point of the compasses on the point A and set its drawing end to point B. The compasses are now set to the length of the sides of the finished triangle. Do not change it from now on. With the compasses' point on P, make two arcs, each roughly where the other two vertices of the triangle will be. On one of the arcs, mark a point Q that will be a second vertex of the triangle. It does not matter which arc you pick, or where on the arc you draw the point. Place the compasses' point on Q and draw an arc that crosses the other arc, creating point R. Using the straightedge, draw three lines linking the points P,Q and R. Done. The triangle PQR is an equilateral triangle. Its side length is equal to the distance AB. You Try! Construct an equilateral triangle with edge length as shown. 9

Construct an square with edge length as shown. 10 11 Bounded Book Construct a square We start with a given line segment AB, which will become one side of the square, and extend line AB to the right. Set the compasses on B and any convenient width. Scribe an arc on each side of B, creating the two points F and G. With the compasses on G and any convenient width, draw an arc above the point B. Without changing the compasses' width, place the compasses on F and draw an arc above B, crossing the previous arc, and creating point H. Draw a line from B through H. This line is perpendicular to AB, so the angle ABH is a right angle (90°); This will become the second side of the square. Set the compasses on A and set its width to AB. This width will be held unchanged as we create the square's other three sides. Draw an arc above point A. Without changing the width, move the compasses to point B. Draw an arc across BH creating point C - a vertex of the square. Without changing the width, move the compasses to C. Draw an arc to the left of C across the exiting arc, creating point D - a vertex of the square. Draw the lines CD and AD. ABCD is a square where each side has a length AB. You Try! Construct an square with edge length as shown.

12 Bounded Book Construct a regular hexagon inscribed in a square Starting with the given circle, center O, mark a point anywhere on the circle. This will be the first vertex of the hexagon. Set the compasses on this point and set the width of the compasses to the center of the circle. The compasses are now set to the radius of the circle. Make an arc across the circle. This will be the next vertex of the hexagon. (It turns out that the side length of a hexagon is equal to its circumradius – the distance from the center to a vertex). Move the compasses on to the next vertex and draw another arc. This is the third vertex of the hexagon. Continue in this way until you have all six vertices. Draw a line between each successive pairs of vertices, for a total of six lines. These lines form a regular hexagon inscribed in the given circle. You Try! Construct a regular hexagon inscribed in a square. 12

Student Printable

Constructions 8 1 Standard: MCC9-12.G.CO.12 make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc). Copying a segment; copying an angle, bisecting a segment; bisecting an angle, constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. MCC9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Essential Question: How do I construct geometric figures? 8 1

2 7 Construct a perpendicular bisector of a line segment You Try!  Place the compasses on one end of the line segment.  Set the compasses' width to a approximately two thirds the line length. The actual width does not matter. Without changing the compasses' width, draw an arc above and below the line.  Again without changing the compasses' width, place the compasses' point on the other end of the line. Draw an arc above and below the line so that the arcs cross the first two.  Using a straightedge, draw a line between the points where the arcs intersect. Done. This line is perpendicular to the first line and bisects it (cuts it at the exact midpoint of the line). You Try! Construct the perpendicular bisector of the line segment. 2 7

12 3 Construct a regular hexagon inscribed in a square You Try! Starting with the given circle, center O, mark a point anywhere on the circle. This will be the first vertex of the hexagon. Set the compasses on this point and set the width of the compasses to the center of the circle. The compasses are now set to the radius of the circle. Make an arc across the circle. This will be the next vertex of the hexagon. (It turns out that the side length of a hexagon is equal to its circumradius – the distance from the center to a vertex). Move the compasses on to the next vertex and draw another arc. This is the third vertex of the hexagon. Continue in this way until you have all six vertices. Draw a line between each successive pairs of vertices, for a total of six lines. These lines form a regular hexagon inscribed in the given circle. You Try! Construct a regular hexagon inscribed in a square. 12 3

Construct an square with edge length as shown. Without changing the width, move the compasses to point B. Draw an arc across BH creating point C - a vertex of the square. Without changing the width, move the compasses to C. Draw an arc to the left of C across the exiting arc, creating point D - a vertex of the square. Draw the lines CD and AD. ABCD is a square where each side has a length AB. You Try! Construct an square with edge length as shown. 4 11

Construct a square We start with a given line segment AB, which will become one side of the square, and extend line AB to the right. Set the compasses on B and any convenient width. Scribe an arc on each side of B, creating the two points F and G. With the compasses on G and any convenient width, draw an arc above the point B. Without changing the compasses' width, place the compasses on F and draw an arc above B, crossing the previous arc, and creating point H. Draw a line from B through H. This line is perpendicular to AB, so the angle ABH is a right angle (90°); This will become the second side of the square. Set the compasses on A and set its width to AB. This width will be held unchanged as we create the square's other three sides. Draw an arc above point A. 10 5

6 9 Construct an equilateral triangle You Try! Starting with the line segment AB which is the length of the sides of the desired equilateral triangle, pick a point P that will be one vertex of the finished triangle. Place the point of the compasses on the point A and set its drawing end to point B. The compasses are now set to the length of the sides of the finished triangle. Do not change it from now on. With the compasses' point on P, make two arcs, each roughly where the other two vertices of the triangle will be. On one of the arcs, mark a point Q that will be a second vertex of the triangle. It does not matter which arc you pick, or where on the arc you draw the point. Place the compasses' point on Q and draw an arc that crosses the other arc, creating point R. Using the straightedge, draw three lines linking the points P,Q and R. Done. The triangle PQR is an equilateral triangle. Its side length is equal to the distance AB. You Try! Construct an equilateral triangle with edge length as shown. 6 9