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10.1: Triangles and Parallelograms

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Understand the basic concepts of area.
  • Use formulas to find the area of triangles and parallelograms.

Review Queue

  1. Define perimeter and area, in your own words.
  2. Solve the equations below. Simplify any radicals.
    1. \begin{align*}x^2=121\end{align*}x2=121
    2. \begin{align*}4x+6=80\end{align*}4x+6=80
    3. \begin{align*}x^2-6x+8=0\end{align*}x26x+8=0
    4. \begin{align*}\frac{1}{2} x-3=5\end{align*}12x3=5
    5. \begin{align*}x^2+2x-15=0\end{align*}x2+2x15=0
    6. \begin{align*}x^2-x-12=0\end{align*}x2x12=0

Know What? Ed’s parents are getting him a new king bed. Upon further research, Ed discovered there are two types of king beds, and Eastern (or standard) King and a California King. The Eastern King has \begin{align*}76” \times 80”\end{align*}76×80 dimensions, while the California King is \begin{align*}72” \times 84”\end{align*}72×84 (both dimensions are \begin{align*}width \times length\end{align*}width×length). Which bed has a larger area to lie on?

Areas and Perimeters of Squares and Rectangles

Perimeter: The distance around a shape.

The perimeter of any figure must have a unit of measurement attached to it. If no specific units are given (feet, inches, centimeters, etc), write “units.”

Example 1: Find the perimeter of the figure to the left.

Solution: Here, we can use the grid as our units. Count around the figure to find the perimeter.

\begin{align*}5 + 1 + 1 + 1 + 5 + 1 + 3 + 1 + 1 + 1 + 1 + 2 + 4 + 7 = 34 \ units\end{align*}5+1+1+1+5+1+3+1+1+1+1+2+4+7=34 units

You are probably familiar with the area of squares and rectangles from a previous math class. Recall that you must always establish a unit of measure for area. Area is always measured in square units, square feet \begin{align*}(ft.^2)\end{align*}(ft.2), square inches \begin{align*}(in.^2)\end{align*}(in.2). square centimeters \begin{align*}(cm.^2)\end{align*}(cm.2), etc. If no specific units are given, write \begin{align*}“units^2.”\end{align*}units2.

Example 2: Find the area of the figure from Example 1.

Solution: Count the number of squares within the figure. If we start on the left and count each column. \begin{align*}5 + 6 + 1 + 4 + 3 + 4 + 4 = 27 \ units^2\end{align*}5+6+1+4+3+4+4=27 units2

Area of a Rectangle: \begin{align*}A=bh\end{align*}, where \begin{align*}b\end{align*} is the base (width) and \begin{align*}h\end{align*} is the height (length).

Example 3: Find the area and perimeter of a rectangle with sides 4 cm by 9 cm.

Solution: The perimeter is \begin{align*}4 + 9 + 4 + 9 = 36 \ cm\end{align*}. The area is \begin{align*}A=9 \cdot 4=26 \ cm^2\end{align*}.

Perimeter of a Rectangle: \begin{align*}P=2b+2h\end{align*}.

If a rectangle is a square, with sides of length \begin{align*}s\end{align*}, the formulas are as follows:

Perimeter of a Square: \begin{align*}P_{square}=2s+2s=4s\end{align*}

Area of a Square: \begin{align*}A_{sqaure}=s \cdot s=s^2\end{align*}

Example 4: The area of a square is \begin{align*}75 \ in^2\end{align*}. Find the perimeter.

Solution: To find the perimeter, we need to find the length of the sides.

\begin{align*}A &= s^2=75 \ in^2\\ s &= \sqrt{75}=5\sqrt{3} \ in\end{align*}

From this, \begin{align*}P=4 \left (5\sqrt{3} \right )=20\sqrt{3} \ in\end{align*}.

Area Postulates

Congruent Areas Postulate: If two figures are congruent, they have the same area.

Example 5: Draw two different rectangles with an area of \begin{align*}36 \ cm^2\end{align*}.

Solution: Think of all the different factors of 36. These can all be dimensions of the different rectangles.

Other possibilities could be \begin{align*}6 \times 6, 2 \times 18\end{align*}, and \begin{align*}1 \times 36\end{align*}.

Example 5 shows two rectangles with the same area and are not congruent. This tells us that the converse of the Congruent Areas Postulate is not true.

Area Addition Postulate: If a figure is composed of two or more parts that do not overlap each other, then the area of the figure is the sum of the areas of the parts.

Example 6: Find the area of the figure below. You may assume all sides are perpendicular.

Solution: Split the shape into two rectangles and find the area of each.

\begin{align*}A_{top \ rectangle} &= 6 \cdot 2=12 \ ft^2\\ A_{bottom \ square} &= 3 \cdot 3=9 \ ft^2\end{align*}

The total area is \begin{align*}12 + 9 = 21 \ ft^2\end{align*}.

Area of a Parallelogram

Recall that a parallelogram is a quadrilateral whose opposite sides are parallel.

To find the area of a parallelogram, make it into a rectangle.

From this, we see that the area of a parallelogram is the same as the area of a rectangle.

Area of a Parallelogram: The area of a parallelogram is \begin{align*}A=bh\end{align*}.

The height of a parallelogram is always perpendicular to the base. This means that the sides are not the height.

Example 7: Find the area of the parallelogram.

Solution: \begin{align*}A=15 \cdot 8=120 \ in^2\end{align*}

Example 8: If the area of a parallelogram is \begin{align*}56 \ units^2\end{align*} and the base is 4 units, what is the height?

Solution: Solve for the height in \begin{align*}A=bh\end{align*}.

\begin{align*}56 &= 4h\\ 14 &= h\end{align*}

Area of a Triangle

If we take parallelogram and cut it in half, along a diagonal, we would have two congruent triangles. The formula for the area of a triangle is half the area of a parallelogram.

Area of a Triangle: \begin{align*}A=\frac{1}{2} \ bh \ \text{or} \ A=\frac{bh}{2}\end{align*}.

Example 9: Find the area of the triangle.

Solution: To find the area, we need to find the height of the triangle. We are given the two sides of the small right triangle, where the hypotenuse is also the short side of the obtuse triangle.

\begin{align*}3^2+h^2 &= 5^2\\ 9+h^2 &= 25\\ h^2 &= 16\\ h &= 4\\ A &= \frac{1}{2} (4)(7)=14 \ units^2\end{align*}

Example 10: Find the perimeter of the triangle in Example 9.

Solution: To find the perimeter, we need to find the longest side of the obtuse triangle. If we used the black lines in the picture, we would see that the longest side is also the hypotenuse of the right triangle with legs 4 and 10.

\begin{align*}4^2+10^2 &= c^2\\ 16+100 &= c^2\\ c &= \sqrt{116} \approx 10.77\end{align*} The perimeter is \begin{align*}7 + 5 + 10. 77 = 22.77 \ units\end{align*}

Example 11: Find the area of the figure below.

Solution: Divide the figure into a triangle and a rectangle with a small rectangle cut out of the lower right-hand corner.

\begin{align*}A &= A_{top \ triangle}+A_{rectangle}-A_{small \ triangle}\\ A &= \left(\frac{1}{2} \cdot 6 \cdot 9\right)+(9 \cdot 15)+\left(\frac{1}{2} \cdot 3 \cdot 6\right)\\ A &= 27+135+9\\ A &= 171 \ units^2\end{align*}

Know What? Revisited The area of an Eastern King is \begin{align*}6080 \ in^2\end{align*} and the California King is \begin{align*}6048 \ in^2\end{align*}.

Review Questions

  • Questions 1-12 are similar to Examples 3-5, 7-9.
  • Questions 13-18 are similar to Examples 9 and 10.
  • Questions 19-24 are similar to Examples 7 and 9.
  • Questions 25-30 are similar to Examples 6 and 11.
  • Questions 31-36 use the formula for the area of a triangle.
  1. Find the area and perimeter of a square with sides of length 12 in.
  2. Find the area and perimeter of a rectangle with height of 9 cm and base of 16 cm.
  3. Find the area of a parallelogram with height of 20 m and base of 18 m.
  4. Find the area and perimeter of a rectangle if the height is 8 and the base is 14.
  5. Find the area and perimeter of a square if the sides are 18 ft.
  6. If the area of a square is \begin{align*}81 \ ft^2\end{align*}, find the perimeter.
  7. If the perimeter of a square is 24 in, find the area.
  8. Find the area of a triangle with base of length 28 cm and height of 15 cm.
  9. What is the height of a triangle with area \begin{align*}144 \ m^2\end{align*} and a base of 24 m?
  10. The perimeter of a rectangle is 32. Find two different dimensions that the rectangle could be.
  11. Draw two different rectangles that haven an area of \begin{align*}90 \ mm^2\end{align*}.
  12. Write the converse of the Congruent Areas Postulate. Determine if it is a true statement. If not, write a counterexample. If it is true, explain why.

Use the triangle to answer the following questions.

  1. Find the height of the triangle by using the geometric mean.
  2. Find the perimeter.
  3. Find the area.

Use the triangle to answer the following questions.

  1. Find the height of the triangle.
  2. Find the perimeter.
  3. Find the area.

Find the area of the following shapes.

    1. Divide the shape into two triangles and one rectangle.
    2. Find the area of the two triangles and rectangle.
    3. Find the area of the entire shape.
    1. Divide the shape into two rectangles and one triangle.
    2. Find the area of the two rectangles and triangle.
    3. Find the area of the entire shape (you will need to subtract the area of the small triangle in the lower right-hand corner).

Use the picture below for questions 27-30. Both figures are squares.

  1. Find the area of the outer square.
  2. Find the area of one grey triangle.
  3. Find the area of all four grey triangles.
  4. Find the area of the inner square.

In questions 31-36 we are going to derive a formula for the area of an equilateral triangle.

  1. What kind of triangle is \begin{align*}\triangle ABD\end{align*}? Find \begin{align*}AD\end{align*} and \begin{align*}BD\end{align*}.
  2. Find the area of \begin{align*}\triangle ABC\end{align*}.
  3. If each side is \begin{align*}x\end{align*}, what is \begin{align*}AD\end{align*} and \begin{align*}BD\end{align*}?
  4. If each side is \begin{align*}x\end{align*}, find the area of \begin{align*}\triangle ABC\end{align*}.
  5. Using your formula from #34, find the area of an equilateral triangle with 12 inch sides.
  6. Using your formula from #34, find the area of an equilateral triangle with 5 inch sides.

Review Queue Answers

1. Possible Answers

Perimeter: The distance around a shape.

Area: The space inside a shape.

2. (a) \begin{align*}x = \pm 11\end{align*}

(b) \begin{align*}x = 18.5\end{align*}

(c) \begin{align*}x = 4,2\end{align*}

(d) \begin{align*}x = 16\end{align*}

(e) \begin{align*}x = 3,-5\end{align*}

(f) \begin{align*}x = 4,-3\end{align*}

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