Q34. - Geometry

Question

a. Think of a parallelogram made with cardboard strips and hinged at each vertex so that the measure of $\angle C$ will vary. Find the area of the parallelogram for each measure of $\angle C$ given30 .

b. Think of a parallelogram made with cardboard strips and hinged at each vertex so that the measure of $\angle C$ will vary. Find the area of the parallelogram for each measure of $\angle C$
given 45.

c. Think of a parallelogram made with cardboard strips and hinged at each vertex so that the measure of  $\angle C$will vary. Find the area of the parallelogram for each measure of$\angle C$  given 60.

d. Think of a parallelogram made with cardboard strips and hinged at each vertex so that the measure of $\angle C$ will vary. Find the area of the parallelogram for each measure of $\angle C$ given 90.

e. Think of a parallelogram made with cardboard strips and hinged at each vertex so that the measure of $\angle C$ will vary. Find the area of the parallelogram for each measure of$\angle C$  given120 .

Approximate your answers to parts (b), (c) and (e) using $\sqrt{2}\approx 1.4$ and$\sqrt{3}\approx 1.7$ . Then record your answers to parts (a)-(c) on a set of axes like the one below.

Step-by-Step Solution

Answer

a. The area of parallelogram is 100.

b. The area of parallelogram is$100\sqrt{2}$ .

c. The area of parallelogram is$100\sqrt{3}$ .

d. The area of parallelogram is 200.

e. The area of parallelogram is$100\sqrt{3}$ .

f. 

The approximate value from part (b) is140 , approximate value from part (c) is170 and

approximate value from part (e) is170 .

Step 1. Analyze the given information.

The measure of $\angle C={30}^{\circ }$

The base of the parallelogram is $b=20$

The Height of the parallelogram is 

$\begin{array}{c}h=10\sin C\\ =10\sin 30\\ =10\cdot \frac{1}{2}\\ =5\end{array}$

Step 2. Concept Used.

Area of parallelogram can be found using the formula

$A=bh$

Where b=base of parallelogram andh= height of parallelogram. 

Step 3. Find the area of parallelogram.

$\begin{array}{c}A=bh\\ =20\times 5\\ =100\end{array}$

Therefore, the area of parallelogram is100.

Step 1. Analyze the given information.

The measure of $\angle C={45}^{\circ }$

The base of the parallelogram is $b=20$

The Height of the parallelogram is 

$\begin{array}{c}h=10\sin C\\ =10\sin 45\\ =10\cdot \frac{1}{\sqrt{2}}\\ =5\sqrt{2}\end{array}$

Step 2. Concept Used.

Area of parallelogram can be found using the formula

$A=bh$

Where b=base of parallelogram and h=height of parallelogram. 

Step 3. Find the area of parallelogram.

$\begin{array}{c}A=bh\\ =20\times 5\sqrt{2}\\ =100\sqrt{2}\end{array}$

Therefore, the area of parallelogram is$100\sqrt{2}$ .

Step 1. Analyze the given information.

The measure of $\angle C={60}^{\circ }$

The base of the parallelogram is b=20

The Height of the parallelogram is 

$\begin{array}{c}h=10\sin C\\ =10\sin 60\\ =10\cdot \frac{\sqrt{3}}{2}\\ =5\sqrt{3}\end{array}$

Step 2. Concept Used.

Area of parallelogram can be found using the formula

$A=bh$

Where b=base of parallelogram and h=height of parallelogram. 

Step 3. Find the area of parallelogram.

$\begin{array}{c}A=bh\\ =20\times 5\sqrt{3}\\ =100\sqrt{3}\end{array}$

Therefore, the area of parallelogram is$100\sqrt{3}$ .

Step 1. Analyze the given information.

The measure of $\angle C={90}^{\circ }$

The base of the parallelogram is b=20

The Height of the parallelogram is 

$\begin{array}{c}h=10\sin C\\ =10\sin 90\\ =10\cdot 1\\ =10\end{array}$

Step 2. Concept Used.

Area of parallelogram can be found using the formula

$A=bh$

Where b=base of parallelogram andh= height of parallelogram. 

Step 3. Find the area of parallelogram.

$\begin{array}{c}A=bh\\ =20\times 10\\ =200\end{array}$

Therefore, the area of parallelogram is200 .

Step 1. Analyze the given information.

The measure of $\angle C={120}^{\circ }$

The base of the parallelogram is b=20

The Height of the parallelogram is 

$\begin{array}{c}h=10\sin C\\ =10\sin 120°\\ =10\sin (90°+30°)\\ =10\cos 30°\\ =10\cdot \frac{\sqrt{3}}{2}=5\sqrt{3}\end{array}$

Step 2. Concept Used.

Area of parallelogram can be found using the formula

$A=bh$

Where b=base of parallelogram andh= height of parallelogram. 

Step 3. Find the area of parallelogram.

$\begin{array}{c}A=bh\\ =20\times 5\sqrt{3}\\ =100\sqrt{3}\end{array}$

Therefore, the area of parallelogram is$100\sqrt{3}$ .

Step 1. Approximate the answers from part (b).

$\begin{array}{c}A=100\sqrt{2}\\ \approx 100\times 1.4\\ =140\end{array}$

Step 2. Approximate the answers from part (c).

$\begin{array}{c}A=100\sqrt{3}\\ \approx 100\times 1.7\\ =170\end{array}$

Step 3. Approximate the answers from part (e).

$\begin{array}{c}A=100\sqrt{3}\\ \approx 100\times 1.7\\ =170\end{array}$

Step 4. Represent the information on axes.

Therefore, the approximate value from part (b) is140 , approximate value from part (c) is170 and approximate value from part (e) is170.