Question

in exercises 33 - 36, draw the polygons described. if there is more than one (or no) shape that you can draw, explain how you know that. 33. draw a rectangle. perimeter = 24 cm and side of 8 cm. 34. draw a triangle. side $overline{ab}=2$ in. side $overline{ac}=1$ in. $angle bac = 75^{circ}$. 35. draw a triangle. $angle bac = 75^{circ}$ and $angle acb = 75^{circ}$. 36. draw a trapezoid $pqrs$. $angle qps = 45^{circ}$. $angle rqp = 45^{circ}$. side $overline{ps}=1$ in. side $overline{pq}=2$ in.

Explanation:

Step1: Analyze rectangle problem

Let the length of the rectangle be $l$ and the width be $w$. The perimeter formula is $P = 2(l + w)$. Given $P=24$ cm and assume $l = 8$ cm. Substitute into the formula: $24=2(8 + w)$.

Step2: Solve for the other side

First, divide both sides of the equation $24 = 2(8 + w)$ by 2: $\frac{24}{2}=8 + w$, so $12=8 + w$. Then subtract 8 from both sides: $w=12 - 8=4$ cm. To draw the rectangle, draw a four - sided figure with opposite sides equal, two sides of length 8 cm and two sides of length 4 cm and all angles equal to 90 degrees.

Step3: Analyze triangle in 34

Use the Side - Angle - Side (SAS) criterion. First draw a ray. Mark a point $A$. Using a ruler, measure and mark point $B$ such that $AB = 2$ in. Then, using a protractor at point $A$, measure an angle of $75^{\circ}$. Along the ray of the $75^{\circ}$ angle, measure and mark point $C$ such that $AC = 1$ in. Connect points $B$ and $C$ to form $\triangle ABC$.

Step4: Analyze triangle in 35

Since the sum of angles in a triangle is $180^{\circ}$, if $\angle BAC=75^{\circ}$ and $\angle ACB = 75^{\circ}$, then $\angle ABC=180-(75 + 75)=30^{\circ}$. But there are infinitely many non - congruent triangles with these angle measures (similar triangles). We can start by drawing a ray. Mark a point $A$. Using a protractor, measure an angle of $75^{\circ}$ at $A$. Then, on the other ray of the $75^{\circ}$ angle, we can choose any point $C$. Then measure an angle of $75^{\circ}$ at $C$ and let the two non - $AC$ rays intersect at point $B$.

Step5: Analyze trapezoid in 36

Draw a ray. Mark point $P$. Using a protractor, measure an angle of $45^{\circ}$ at $P$ and draw a ray. Mark point $Q$ on this ray such that $PQ = 2$ in. Then, at point $Q$, measure an angle of $45^{\circ}$ in the appropriate direction. At point $P$, measure a line segment $PS = 1$ in along a ray. Draw a line parallel to $PQ$ from point $S$ that intersects the ray from $Q$ at point $R$ to form trapezoid $PQRS$.

Answer:

• For 33: Draw a rectangle with sides 8 cm and 4 cm.
• For 34: Use SAS to draw $\triangle ABC$ with $AB = 2$ in, $\angle BAC=75^{\circ}$, $AC = 1$ in.
• For 35: Infinitely many non - congruent triangles can be drawn as there are no side - length constraints, only angle constraints.
• For 36: Draw trapezoid $PQRS$ with $\angle QPS = 45^{\circ}$, $\angle RQP = 45^{\circ}$, $PS = 1$ in, $PQ = 2$ in.