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Chapter 1.6 ConstructionsI can make basic constructions using a straightedge
and a compass
Success Criteria: I can use special geometric tools to
make a figure that is congruent to an original figure without measuring
I can apply this method that is more accurate than measuring
Today1. Do Now 2. Lesson 1.63. Check HW #6
Do NowFind the following measures using the image.
DefinitionsConstruction – Use a straight edge and a
compass to make geometric figures
Straightedge – is a ruler with no marking on it
Compass – Geometric tool used to draw circles and parts of circles called arcs
Perpendicular Lines-two lines that intersect to form right angles.
Perpendicular bisector-of a segment is a line that is perpendicular to the midpoint
Congruent segment
http://www.mathopenref.com/constcopysegment.html
Congruent Angles
http://www.mathopenref.com/constcopyangle.html
Perpendicular Bisector
http://www.mathopenref.com/constbisectline.html
Angle Bisector
http://www.mathopenref.com/constbisectangle.html
Construct a trianglegiven three line segments
http://www.mathopenref.com/consttrianglesss.html
Step 2: Open the compass to the length of KM.
Construct TW congruent to KM.
Step 1: Draw a ray with endpoint T.
Step 3: With the same compass setting, put the compass point on point T. Draw an arc that intersects the ray. Label the point of intersection W.
TW KM
Construct Y so that Y G.
Step 1: Draw a ray with endpoint Y.
Step 3: With the same compass setting, put the compass point on point Y. Draw an arc that intersects the ray. Label the point of intersection Z.
Step 2: With the compass point on point G, draw an arc that intersects both sides of G. Label the points of intersection E and F.
75°
(continued)
Step 4: Open the compass to the length EF. Keeping the same compass setting, put the compass point on Z. Draw an arc that intersects the arc you drew in Step 3. Label the point of intersection X.
Y G
Step 5: Draw YX to complete Y.
Start with AB.
Step 2: With the same compass setting, put the compass point on point B and draw a short arc.
Without two points of intersection, no line can be drawn, so the perpendicular bisector cannot be drawn.
Prove by construction why you cannot construct a
perpendicular bisector with a compass opening less than AB. 12
Step 1: Put the compass point on
point A and draw a short arc. Make
sure that the opening is less than AB.12
–3x = –48 Subtract 4x from each side. x = 16 Divide each side by –3.
m AWR = m BWR Definition of angle bisector x = 4x – 48 Substitute x for m AWR and
4x – 48 for m BWR.
m AWB = m AWR + m BWR Angle Addition Postulatem AWB = 16 + 16 = 32 Substitute 16 for m AWR and
for m BWR.
Draw and label a figure to illustrate the problem
WR bisects AWB. m AWR = x and m BWR = 4x – 48. Find m AWB.
m AWR = 16 m BWR = 4(16) – 48 = 16 Substitute 16 for x.
Step 1: Put the compass point on vertex M. Draw an arc that intersects both sides of M. Label the points of intersection B and C.
Step 2: Put the compass point on point B. Draw an arc in the interior of M.
Construct MX, the bisector of M.
Step 4: Draw MX. MX is the angle bisector of M.
(continued)
Step 3: Put the compass point on point C. Using the same compass setting, draw an arc in the interior of M. Make sure that the arcs intersect. Label the point where the two arcs intersect X.
You can make a sketch or measure and draw a segment. These may not be exact. A construction is a way of creating a figure that is more precise. One way to make a geometric construction is to use a compass and straightedge.
Example 2: Copying a Segment
Sketch, draw, and construct a segment congruent to MN.
Step 1 Estimate and sketch. Estimate the length of MN and sketch PQ approximately the same length.
P Q
Example 2 Continued
Sketch, draw, and construct a segment congruent to MN.
Step 2 Measure and draw. Use a ruler to measure MN. MN appears to be 3.5 in. Use a ruler to draw XY to have length 3.5 in.
X Y
Example 2 Continued
Sketch, draw, and construct a segment congruent to MN.
Step 3 Construct and compare. Use a compass and straightedge to construct ST congruent to MN.
A ruler shows that PQ and XY are approximately the same length as MN, but ST is precisely the same length.
Check It Out! Example 2
Sketch, draw, and construct a segment congruent to JK.
Step 1 Estimate and sketch. Estimate the length of JK and sketch PQ approximately the same length.
Check It Out! Example 2 Continued
Step 2 Measure and draw. Use a ruler to measure JK. JK appears to be 1.7 in. Use a ruler to draw XY to have length 1.7 in.
Sketch, draw, and construct a segment congruent to JK.