3. given triangle oab shown below with points u, r, s, and t marked, which of the following points is the circumcenter of $ \triangle oab $?
(i) u
(ii) r
(iii) s
(iv) t

3. for equilateral triangle pqr shown below, do the following:
(a) construct the circle that circumscribes $ \triangle pqr $. label the circumcenter as point m.
(b) what is the minimum angle of rotation about point m that will map $ \triangle pqr $ onto itself? show how you found your answer.

4. for isosceles triangle abc shown below, $ \overline{ab} \cong \overline{ac} $.
(a) construct the circumcenter for $ \triangle abc $. label it as point d. leave all construction marks.
(b) draw in the circumcircle for $ \triangle abc $.
(c) draw in $ \overline{ad} $, $ \overline{cd} $, and $ \overline{bd} $.

reasoning
5. give an explanation why $ \triangle acd $ must be congruent to $ \triangle abd $ in the diagram above.

Question

3. given triangle oab shown below with points u, r, s, and t marked, which of the following points is the circumcenter of $ \triangle oab $?
(i) u
(ii) r
(iii) s
(iv) t

3. for equilateral triangle pqr shown below, do the following:
(a) construct the circle that circumscribes $ \triangle pqr $. label the circumcenter as point m.
(b) what is the minimum angle of rotation about point m that will map $ \triangle pqr $ onto itself? show how you found your answer.

4. for isosceles triangle abc shown below, $ \overline{ab} \cong \overline{ac} $.
(a) construct the circumcenter for $ \triangle abc $. label it as point d. leave all construction marks.
(b) draw in the circumcircle for $ \triangle abc $.
(c) draw in $ \overline{ad} $, $ \overline{cd} $, and $ \overline{bd} $.

reasoning
5. give an explanation why $ \triangle acd $ must be congruent to $ \triangle abd $ in the diagram above.

Accepted Answer

### For Question 5 (Explaining why $\boldsymbol{	riangle ACD \cong 	riangle ABD}$):

# Brief Explanations:
1. **Given Information**: $	riangle ABC$ is isosceles with $AB = AC$ (from the problem statement: $\overline{AB} \cong \overline{AC}$).
2. **Circumcenter Property**: Point $D$ is the circumcenter of $	riangle ABC$, so $D$ is equidistant from all vertices: $AD = AD$ (common side), $CD = BD$ (radii of the circumcircle), and $AC = AB$ (given).
3. **SSS Congruence Criterion**: By the Side - Side - Side (SSS) congruence rule, if three sides of one triangle are equal to three corresponding sides of another triangle, the triangles are congruent. Here, $AC = AB$, $CD = BD$, and $AD = AD$, so $	riangle ACD \cong 	riangle ABD$ by SSS.

# Answer:
$	riangle ACD \cong 	riangle ABD$ by the SSS (Side - Side - Side) congruence criterion. This is because:
- $AC = AB$ (given, as $	riangle ABC$ is isosceles with $\overline{AB} \cong \overline{AC}$).
- $CD = BD$ (radii of the circumcircle of $	riangle ABC$, since $D$ is the circumcenter).
- $AD = AD$ (common side to both $	riangle ACD$ and $	riangle ABD$).

Thus, by SSS, $	riangle ACD \cong 	riangle ABD$.

### For Question 3 (Equilateral Triangle $\boldsymbol{	riangle PQR}$):

#### Part (a): Constructing the Circumcircle of $\boldsymbol{	riangle PQR}$
# Brief Explanations:
1. **Circumcenter Definition**: The circumcenter of a triangle is the intersection of its perpendicular bisectors. For an equilateral triangle, the circumcenter, centroid, incenter, and orthocenter coincide.
2. **Steps to Construct**:
   - Draw the perpendicular bisector of one side (e.g., $RQ$):
     - Find the midpoint of $RQ$ (let's call it $M$) by measuring the length of $RQ$ and dividing by 2, or by using a compass to draw arcs from $R$ and $Q$ with equal radii and finding their intersection points, then drawing the line through these points (the perpendicular bisector).
   - Repeat the process for another side (e.g., $PR$) to find its midpoint $N$ and draw its perpendicular bisector.
   - The intersection of these two perpendicular bisectors is the circumcenter $M$ (for an equilateral triangle, this point is also the centroid, incenter, etc.).
   - With $M$ as the center and $MR$ (or $MQ$ or $MP$) as the radius, draw the circle. This circle is the circumcircle of $	riangle PQR$.

#### Part (b): Minimum Angle of Rotation about $\boldsymbol{M}$ to Map $\boldsymbol{	riangle PQR}$ onto Itself
# Explanation:
## Step 1: Recall Rotational Symmetry of Equilateral Triangle
An equilateral triangle has rotational symmetry of order 3. This means it can be rotated about its center (circumcenter $M$) by angles that are multiples of $\frac{360^{\circ}}{3}=120^{\circ}$ and still map onto itself.

## Step 2: Determine Minimum Angle
The smallest positive angle of rotation is the smallest angle in this set of multiples. The multiples are $120^{\circ}, 240^{\circ}, 360^{\circ}$, etc. The smallest positive angle is $120^{\circ}$.

# Answer:
(a) The circumcircle is constructed by finding the perpendicular bisectors of two sides of $	riangle PQR$ (they intersect at the circumcenter $M$), then drawing a circle with center $M$ and radius equal to the distance from $M$ to any vertex (e.g., $MR$).

(b) The minimum angle of rotation about point $M$ that maps $	riangle PQR$ onto itself is $120^{\circ}$. This is because an equilateral triangle has rotational symmetry of order 3, so the angle of rotation is $\frac{360^{\circ}}{3}=120^{\circ}$.

### For Question 4 (Isosceles Triangle $\boldsymbol{	riangle ABC}$ with $\boldsymbol{AB = AC}$):

#### Part (a): Constructing the Circumcenter $\boldsymbol{D}$ of $\boldsymbol{	riangle ABC}$
# Brief Explanations:
1. **Circumcenter Definition**: The circumcenter is the intersection of the perpendicular bisectors of the triangle's sides. For an isosceles triangle with $AB = AC$, the perpendicular bisector of the base $BC$ will also be the altitude, median, and angle bisector from $A$.
2. **Steps to Construct**:
   - Draw the perpendicular bisector of $BC$:
     - Find the midpoint of $BC$ (let's call it $E$) by measuring $BC$ and dividing by 2, or by using a compass to draw arcs from $B$ and $C$ with equal radii, finding the intersection points, and drawing the line through them (perpendicular bisector of $BC$).
   - Draw the perpendicular bisector of $AB$ (or $AC$):
     - Find the midpoint of $AB$ (let's call it $F$) and draw the perpendicular bisector (using the same arc - drawing method as above).
   - The intersection of these two perpendicular bisectors is the circumcenter $D$. (For an isosceles triangle, the perpendicular bisector of the base $BC$ and the perpendicular bisector of $AB$ (or $AC$) will intersect at $D$.)

#### Part (b): Drawing the Circumcircle of $\boldsymbol{	riangle ABC}$
# Brief Explanations:
With $D$ as the center (circumcenter) and $DB$ (or $DC$ or $DA$) as the radius, draw a circle. This circle will pass through all three vertices $A$, $B$, and $C$, so it is the circumcircle of $	riangle ABC$.

#### Part (c): Drawing $\boldsymbol{\overline{AD}}$, $\boldsymbol{\overline{CD}}$, and $\boldsymbol{\overline{BD}}$
# Brief Explanations:
- $\overline{AD}$: Draw a line segment from vertex $A$ to the circumcenter $D$.
- $\overline{CD}$: Draw a line segment from vertex $C$ to the circumcenter $D$.
- $\overline{BD}$: Draw a line segment from vertex $B$ to the circumcenter $D$.

These segments are the radii of the circumcircle (since $D$ is the circumcenter, $AD = BD = CD$).

### For Question 2 (Choices about the Triangle with Points $\boldsymbol{O, A, B}$ etc. - Assuming a Typical Triangle Problem):
(Note: Since the exact question text is a bit unclear, but if it's about identifying a point or a length related to a triangle with points $O, A, B$ and a triangle with sides labeled, for example, if it's about the circumradius or a segment length, we can assume the following general approach.)

# Brief Explanations:
If it's a multiple - choice question about which point (e.g., $O$, $A$, $B$, $P$) is the circumcenter or related to a segment length:
- For a triangle, the circumcenter is equidistant from all vertices. If the triangle has a right angle, the circumcenter is at the midpoint of the hypotenuse. If it's an isosceles or equilateral triangle, use the properties of those triangles (perpendicular bisectors, midpoints, etc.) to determine the correct point or length.

(If you can provide the exact text of Question 2, a more precise answer can be given.)

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QUESTION IMAGE

Question

Explanation:

For Question 5 (Explaining why $\boldsymbol{\triangle ACD \cong \triangle ABD}$):

Part (b): Minimum Angle of Rotation about $\boldsymbol{M}$ to Map $\boldsymbol{\triangle PQR}$ onto Itself

Step 1: Recall Rotational Symmetry of Equilateral Triangle

Step 2: Determine Minimum Angle

Answer:

For Question 3 (Equilateral Triangle $\boldsymbol{\triangle PQR}$):

Part (a): Constructing the Circumcircle of $\boldsymbol{\triangle PQR}$