Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s...
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![Page 1: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/1.jpg)
Step-by-StepConstructions
PerpendicularThrough A Point
(off a line)
EquilateralTriangle
What’s Wrongwith SSA?
Circumcenter
Incenter
A
P2P1
A2A1 P1
B
P4
P3
P2P1
P1 P2
P3
P4
P5
Centroid
P3
P1 P2
P4 PerpendicularBisector
[click on a construction]
![Page 2: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/2.jpg)
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PerpendicularBisector
![Page 3: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/3.jpg)
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1. adjust compass setting so that it is over half the length of given segment
2. construct an arc from each endpoint of the segment, P1 and P2, (arcs must intersect one another twice)
3. connect the 2 points where the arcs intersect, A and B (this segment, AB, is the perpendicular bisector).
The point where this perpendicular bisector intersects the original segment, C, is the midpoint
P1
Perpendicular Bisector
![Page 4: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/4.jpg)
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1. adjust compass setting so that it is over half the length of given segment
2. construct an arc from each endpoint of the segment, P1 and P2, (arcs must intersect one another twice)
3. connect the 2 points where the arcs intersect, A and B (this segment, AB, is the perpendicular bisector).
The point where this perpendicular bisector intersects the original segment, C, is the midpoint
P2P1
Perpendicular Bisector
![Page 5: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/5.jpg)
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1. adjust compass setting so that it is over half the length of given segment
2. construct an arc from each endpoint of the segment, P1 and P2, (arcs must intersect one another twice)
3. connect the 2 points where the arcs intersect, A and B (this segment, AB, is the perpendicular bisector).
The point where this perpendicular bisector intersects the original segment, C, is the midpoint
B
A
CP1 P2
Perpendicular Bisector
![Page 6: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/6.jpg)
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1. adjust compass setting so that it is over half the length of given segment
2. construct an arc from each endpoint of the segment, P1 and P2, (arcs must intersect one another twice)
3. connect the 2 points where the arcs intersect, A and B (this segment, AB, is the perpendicular bisector).
The point where this perpendicular bisector intersects the original segment, C, is the midpoint
B
A
CP1 P2
Perpendicular Bisector
![Page 7: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/7.jpg)
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PerpendicularThrough A
Point(off the line)
A
P1
P3P2
![Page 8: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/8.jpg)
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Perpendicular Through a Point
1. adjust compass setting so that it is slightly more than the distance from the given point, P1, to the line
2. place the compass point on P1 and construct an arc that intersects the line in 2 places, P2 and P3 (if there's not enough room, you may need to extend the original line segment)
3. place the compass point on P2 and construct an arc on the opposite side of the segment from the given point, P1
4. repeat step #3 from other arc-line intersection, P3
5. connect the original point, P1, to the point where these two new arcs intersect, A, to create your perpendicular
P3P2
P1
![Page 9: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/9.jpg)
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Perpendicular Through a Point
1. adjust compass setting so that it is slightly more than the distance from the given point, P1, to the line
2. place the compass point on P1 and construct an arc that intersects the line in 2 places, P2 and P3 (if there's not enough room, you may need to extend the original line segment)
3. place the compass point on P2 and construct an arc on the opposite side of the segment from the given point, P1
4. repeat step #3 from other arc-line intersection, P3
5. connect the original point, P1, to the point where these two new arcs intersect, A, to create your perpendicular
P1
P1
P3P2
![Page 10: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/10.jpg)
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Perpendicular Through a Point
1. adjust compass setting so that it is slightly more than the distance from the given point, P1, to the line
2. place the compass point on P1 and construct an arc that intersects the line in 2 places, P2 and P3 (if there's not enough room, you may need to extend the original line segment)
3. place the compass point on P2 and construct an arc on the opposite side of the segment from the given point, P1
4. repeat step #3 from other arc-line intersection, P3
5. connect the original point, P1, to the point where these two new arcs intersect, A, to create your perpendicular
P2 P3
P1
![Page 11: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/11.jpg)
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Perpendicular Through a Point
1. adjust compass setting so that it is slightly more than the distance from the given point, P1, to the line
2. place the compass point on P1 and construct an arc that intersects the line in 2 places, P2 and P3 (if there's not enough room, you may need to extend the original line segment)
3. place the compass point on P2 and construct an arc on the opposite side of the segment from the given point, P1
4. repeat step #3 from other arc-line intersection, P3
5. connect the original point, P1, to the point where these two new arcs intersect, A, to create your perpendicular
A
P1
P3P2
![Page 12: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/12.jpg)
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Perpendicular Through a Point
1. adjust compass setting so that it is slightly more than the distance from the given point, P1, to the line
2. place the compass point on P1 and construct an arc that intersects the line in 2 places, P2 and P3 (if there's not enough room, you may need to extend the original line segment)
3. place the compass point on P2 and construct an arc on the opposite side of the segment from the given point, P1
4. repeat step #3 from other arc-line intersection, P3
5. connect the original point, P1, to the point where these two new arcs intersect, A, to create your perpendicular
A
P1
P3P2
![Page 13: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/13.jpg)
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Equilateral Triangle
A
P2P1
![Page 14: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/14.jpg)
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Equilateral Triangle
1. construct a segment2. place the compass tip on one
endpoint, P1, and adjust the pencil so that it reaches the other endpoint, P2. Construct an arc above the segment (using the length of the segment as the measure of your compass)
3. repeat step #2 from the other endpoint, P2 (keeping the compass measure the same) to locate A
4. connect the point where these two arcs intersect, A, to create an equilateral triangle
P2P1
![Page 15: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/15.jpg)
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Equilateral Triangle
1. construct a segment2. place the compass tip on one
endpoint, P1, and adjust the pencil so that it reaches the other endpoint, P2. Construct an arc above the segment (using the length of the segment as the measure of your compass)
3. repeat step #2 from the other endpoint, P2 (keeping the compass measure the same) to locate A
4. connect the point where these two arcs intersect, A, to create an equilateral triangle
P1 P2
A
![Page 16: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/16.jpg)
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Equilateral Triangle
1. construct a segment2. place the compass tip on one
endpoint, P1, and adjust the pencil so that it reaches the other endpoint, P2. Construct an arc above the segment (using the length of the segment as the measure of your compass)
3. repeat step #2 from the other endpoint, P2 (keeping the compass measure the same) to locate A
4. connect the point where these two arcs intersect, A, to create an equilateral triangle
A
P2P1
![Page 17: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/17.jpg)
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Equilateral Triangle
1. construct a segment2. place the compass tip on one
endpoint, P1, and adjust the pencil so that it reaches the other endpoint, P2. Construct an arc above the segment (using the length of the segment as the measure of your compass)
3. repeat step #2 from the other endpoint, P2 (keeping the compass measure the same) to locate A
4. connect the point where these two arcs intersect, A, to create an equilateral triangle
A
P2P1
![Page 18: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/18.jpg)
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What’s Wrong with
SSA?A2A1 P1
B
![Page 19: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/19.jpg)
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What’s Wrong with SSA?
B
P1A
1. duplicate one angle at P1
2. duplicate one side, P1B 3. duplicate the next consecutive
side, BA
note, there are two possible intersections for the duplicated triangle P1BA1or2
Original Triangle
Duplication
P1
![Page 20: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/20.jpg)
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What’s Wrong with SSA?
A P1
B
1. duplicate one angle at P1
2. duplicate one side, P1B 3. duplicate the next consecutive
side, BA
note, there are two possible intersections for the duplicated triangle P1BA1or2
Original Triangle
Duplication
B
P1
![Page 21: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/21.jpg)
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What’s Wrong with SSA?
B
P1A
1. duplicate one angle at P1
2. duplicate one side, P1B 3. duplicate the next consecutive
side, BA
note, there are two possible intersections for the duplicated triangle P1BA1or2
Original Triangle
Duplication
A2A1 P1
B
![Page 22: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/22.jpg)
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What’s Wrong with SSA?
A2A1 P1
B
1. duplicate one angle at P1
2. duplicate one side, P1B 3. duplicate the next consecutive
side, BA
note, there are two possible intersections for the duplicated triangle P1BA1or2
![Page 23: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/23.jpg)
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CircumcenterP4
P3
P2P1
![Page 24: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/24.jpg)
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Circumcenter
1. construct at least two perpendicular bisectors of the sides of a triangle (the intersection of the perpendicular bisectors, P4, is the circumcenter)
2. place the compass tip on the circumcenter, P4, and adjust the compass to a vertex, P1 (this is the radius of the circumscribed circle), then construct the circumscribed circle
P4
P1 P2
P3
![Page 25: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/25.jpg)
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Circumcenter
1. construct at least two perpendicular bisectors of the sides of a triangle (the intersection of the perpendicular bisectors, P4, is the circumcenter)
2. place the compass tip on the circumcenter, P4, and adjust the compass to a vertex, P1 (this is the radius of the circumscribed circle), then construct the circumscribed circle
P4
P3
P2P1
![Page 26: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/26.jpg)
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Circumcenter
1. construct at least two perpendicular bisectors of the sides of a triangle (the intersection of the perpendicular bisectors, P4, is the circumcenter)
2. place the compass tip on the circumcenter, P4, and adjust the compass to a vertex, P1 (this is the radius of the circumscribed circle), then construct the circumscribed circle
P4
P3
P2P1
![Page 27: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/27.jpg)
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IncenterP1 P2
P3
P4
P5
![Page 28: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/28.jpg)
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Incenter
1. construct at least two angle bisectors of the triangle (the intersection of the angle bisectors, P4, is the incenter)
2. from the incenter, construct a perpendicular to any side of the triangle (this is the radius of the inscribed circle)
3. place the compass tip on the incenter and adjust the compass to the measure of the perpendicular from step #2, then construct the inscribed circle
P4
P1 P2
P3
![Page 29: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/29.jpg)
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Incenter
1. construct at least two angle bisectors of the triangle (the intersection of the angle bisectors, P4, is the incenter)
2. from the incenter, construct a perpendicular to any side of the triangle (this is the radius of the inscribed circle)
3. place the compass tip on the incenter and adjust the compass to the measure of the perpendicular from step #2, then construct the inscribed circle
P5
P4
P3
P2P1
![Page 30: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/30.jpg)
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Incenter
1. construct at least two angle bisectors of the triangle (the intersection of the angle bisectors, P4, is the incenter)
2. from the incenter, construct a perpendicular to any side of the triangle (this is the radius of the inscribed circle)
3. place the compass tip on the incenter and adjust the compass to the measure of the perpendicular from step #2, then construct the inscribed circle
P1 P2
P3
P4
P5
![Page 31: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/31.jpg)
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Incenter
1. construct at least two angle bisectors of the triangle (the intersection of the angle bisectors, P4, is the incenter)
2. from the incenter, construct a perpendicular to any side of the triangle (this is the radius of the inscribed circle)
3. place the compass tip on the incenter and adjust the compass to the measure of the perpendicular from step #2, then construct the inscribed circle
P1 P2
P3
P4
P5
![Page 32: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/32.jpg)
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CentroidB
A
C
P3
P1 P2
![Page 33: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/33.jpg)
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Centroid
1. construct at least two perpendicular bisectors of the triangle, label the midpoints A and B
2. connect each midpoint, A and B, to its opposite vertex, P3 and P2 to form medians AP3 and BP2. The point of intersection of these two medians is the centroid, C
B
AP1 P2
P3
![Page 34: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/34.jpg)
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Centroid
1. construct at least two perpendicular bisectors of the triangle, label the midpoints A and B
2. connect each midpoint, A and B, to its opposite vertex, P3 and P2 to form medians AP3 and BP2. The point of intersection of these two medians is the centroid, C
B
A
C
P3
P1 P2
![Page 35: Step-by-Step Constructions Perpendicular Through A Point (off a line) Equilateral Triangle What’s Wrong with SSA? Circumcenter Incenter Centroid Perpendicular.](https://reader030.fdocuments.net/reader030/viewer/2022032612/56649eb05503460f94bb5baa/html5/thumbnails/35.jpg)
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Centroid
1. construct at least two perpendicular bisectors of the triangle, label the midpoints A and B
2. connect each midpoint, A and B, to its opposite vertex, P3 and P2 to form medians AP3 and BP2. The point of intersection of these two medians is the centroid, C
B
A
C
P3
P1 P2