Construct a triangle $$ riangle A B C $$ of sides $$ m \overline{A B}=m \overline{B C}=m \overline{A C}=5 \mathrm{~cm} $$ $$ m \overline{A B}=m \overline{B C}=m \overline{A C}=5 \mathrm{~cm} $$ (a) Construct perpendicular bisectors of the triangle. (b) Write down the construction

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Construct a triangle $$ 	riangle A B C $$ of sides $$ m \overline{A B}=m \overline{B C}=m \overline{A C}=5 \mathrm{~cm} $$ $$ m \overline{A B}=m \overline{B C}=m \overline{A C}=5 \mathrm{~cm} $$ (a) Construct perpendicular bisectors of the triangle. (b) Write down the construction

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1. Draw a segment $$ \overline{A B} $$ of length 5 cm using a ruler.

2. With center $$ A $$ and radius 5 cm (using a compass), draw an arc.  
   With center $$ B $$ and radius 5 cm, draw another arc. The two arcs intersect at a point; label this intersection $$ C $$.

3. Connect the points $$ A $$ and $$ C $$ with a straight line, and connect the points $$ B $$ and $$ C $$ with a straight line.  
   This completes the construction of the equilateral triangle $$ \triangle ABC $$, where
   $$
   m\overline{A B} = m\overline{B C} = m\overline{A C} = 5\,\text{cm}.
   $$

4. To construct the perpendicular bisector of side $$ \overline{A B} $$:  
   - With the compass set to a radius greater than half the length of $$ \overline{A B} $$, place the point of the compass on $$ A $$ and draw an arc above and below the segment.  
   - Without changing the compass width, repeat the same from point $$ B $$.  
   - Mark the two intersections of the arcs.  
   - Draw the line connecting these two intersections; this is the perpendicular bisector of $$ \overline{A B} $$.

5. Repeat the procedure in Step 4 for sides $$ \overline{B C} $$ and $$ \overline{A C} $$ to construct their perpendicular bisectors.

6. The intersection of the three perpendicular bisectors is the circumcenter of $$ \triangle ABC $$.

Construction Summary:  
- Draw segment $$ \overline{A B} $$ of 5 cm.  
- Use arcs to locate $$ C $$ such that $$ m\overline{A C} = m\overline{B C} = 5\,\text{cm} $$.  
- Connect $$ A $$, $$ B $$, and $$ C $$ to form the equilateral triangle $$ \triangle ABC $$.  
- For each side, with a compass set to a radius greater than half the side, draw arcs from its endpoints.  
- Connect the intersections of these arcs for each side to form the perpendicular bisectors.

This completes the construction instructions for both the triangle and its perpendicular bisectors.
To construct triangle $$ \triangle ABC $$ with all sides equal to 5 cm: 1. Draw segment $$ \overline{A B} $$ of 5 cm. 2. Using a compass, draw arcs from points $$ A $$ and $$ B $$ with a radius of 5 cm to locate point $$ C $$. 3. Connect points $$ A $$, $$ B $$, and $$ C $$ to form the equilateral triangle. 4. For each side, draw the perpendicular bisector by creating arcs from both endpoints and connecting the intersection points. This results in an equilateral triangle with all sides equal and the perpendicular bisectors constructed for each side.

Construct a triangle \( \triangle A B C \) of sides \( m \overline{A B}=m \overline{B C}=m \overline{A C}=5 \mathrm{~cm} \) \( m \overline{A B}=m \overline{B C}=m \overline{A C}=5 \mathrm{~cm} \) (a) Construct perpendicular bisectors of the triangle. (b) Write down the construction

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