3.4 Solve geometry applications: triangles, rectangles, and the  (Page 6/8)

The length of a rectangle is four centimeters more than twice the width. The perimeter is 32 centimeters. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and the width
Step 3. Name. Choose a variable to represent the width.
The length is four more than twice the width.
Step 4. Translate
Write the appropriate formula.
Substitute in the given information.
Step 5. Solve the equation.	12 The length is 12 cm.
Step 6. Check. $\begin{array}{ccc}\hfill P& =\hfill & 2L+2W\hfill \\ \hfill 32& \stackrel{?}{=}\hfill & 2\cdot 12+2\cdot 4\hfill \\ \hfill 32& =\hfill & 32✓\hfill \end{array}$
Step 7. Answer the question.	The length is 12 cm and the width is 4 cm.

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The perimeter of a rectangular swimming pool is 150 feet. The length is 15 feet more than the width. Find the length and width.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	$P=150$ ft
Step 2. Identify what you are looking for.	the length and the width of the pool
Step 3. Name. Choose a variable to represent the width. The length is 15 feet more than the width.
Step 4. Translate
Write the appropriate formula.
Substitute.
Step 5. Solve the equation.
Step 6. Check. $\begin{array}{ccc}\hfill P& =\hfill & 2L+2W\hfill \\ \hfill 150& \stackrel{?}{=}\hfill & 2\left(45\right)+2\left(30\right)\hfill \\ \hfill 150& =\hfill & 150✓\hfill \end{array}$
Step 7. Answer the question.	The length of the pool is 45 feet and the width is 30 feet.

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Key concepts

• Problem-Solving Strategy for Geometry Applications
  1. Read the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.
  2. Identify what we are looking for.
  3. Name what we are looking for by choosing a variable to represent it.
  4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.
• Triangle Properties For $\text{△}ABC$
  Angle measures:
  
  • $m\text{∠}A+m\text{∠}B+m\text{∠}C=180$
  Perimeter:
  
  • $P=a+b+c$
  Area:
  
  • $A=\frac{1}{2}bh,\text{b}=\text{base},\text{h}=\text{height}$
  A right triangle has one $90\text{°}$ angle.
• The Pythagorean Theorem In any right triangle, $a^2+b^2=c^2$ where c is the length of the hypotenuse and a and b are the lengths of the legs.
• Properties of Rectangles
  • Rectangles have four sides and four right (90°) angles.
  • The lengths of opposite sides are equal.
  • The perimeter of a rectangle is the sum of twice the length and twice the width: $P=2L+2W.$ The area of a rectangle is the length times the width: $A=LW.$