Warm Up

Find the unknown side length in each right triangle with legs a and b and hypotenuse c.

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1. a = 20, b = 21

2. b = 21, c = 35

3. a = 20, c = 52

c = 29

a = 28

b = 48

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Develop and apply the formulas for the areas of triangles and special quadrilaterals.

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Solve problems involving perimeters and areas of triangles and special quadrilaterals.

Objectives

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

A tangram is an ancient Chinese puzzle made from a square. The pieces can be rearranged to form many different shapes. The area of a figure made with all the pieces is the sum of the areas of the pieces.

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Recall that a rectangle with base b and height h has an area of A = bh.

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You can use the Area Addition Postulate to see that a parallelogram has the same area as a rectangle with the same base and height.

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Remember that rectangles and squares are also parallelograms. The area of a square with side s is A = s², and the perimeter is P = 4s.

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Find the area of the parallelogram.

Example 1A: Finding Measurements of Parallelograms

Step 1 Use the Pythagorean Theorem to find the height h.

Step 2 Use h to find the area of the parallelogram.

Simplify.

Substitute 11 for b and 16 for h.

Area of a parallelogram

30² + h² = 34²

h = 16

A = bh

A = 11(16)

A = 176 mm²

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Example 1B: Finding Measurements of Parallelograms

Find the height of a rectangle in which b = 3 in. and A = (6x² + 24x – 6) in².

Sym. Prop. of =

Divide both sides by 3.

Factor 3 out of the expression for A.

Substitute 6x² + 24x – 6 for A and 3 for b.

Area of a rectangle

A = bh

6x² + 24x – 6 = 3h

3(2x² + 8x – 2) = 3h

2x² + 8x – 2 = h

h = (2x² + 8x – 2) in.

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Example 1C: Finding Measurements of Parallelograms

Find the perimeter of the rectangle, in which

A = (79.8x² – 42) cm²

Step 1 Use the area and the height to find the base.

Substitute 79.8x² – 42 for A and 21 for h.

Divide both sides by 21.

Area of a rectangle

A = bh

79.8x² – 42 = b(21)

3.8x² – 2 = b

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Example 1C Continued

Step 2 Use the base and the height to find the perimeter.

Simplify.

Perimeter of a rectangle

P = 2b + 2h

P = 2(3.8x² – 2) + 2(21)

P = (7.6x² + 38) cm

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Check It Out! Example 1

A = bh

Find the base of the parallelogram in which h = 56 yd and A = 28 yd².

28 = b(56)

b = 0.5 yd

Area of a parallelogram

Substitute.

Simplify.

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Find the area of a trapezoid in which b₁ = 8 in.,

b₂ = 5 in., and h = 6.2 in.

Example 2A: Finding Measurements of Triangles and Trapezoids

Simplify.

Area of a trapezoid

Substitute 8 for b₁, 5 for b₂, and 6.2 for h.

A = 40.3 in²

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Example 2B: Finding Measurements of Triangles and Trapezoids

Find the base of the triangle, in which A = (15x²) cm².

Sym. Prop. of =

Divide both sides by x.

Substitute 15x² for A and 5x for h.

Area of a triangle

6x = b

b = 6x cm

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Example 2C: Finding Measurements of Triangles and Trapezoids

Find b₂ of the trapezoid, in which A = 231 mm².

Sym. Prop. of =

Subtract 23 from both sides.

Area of a trapezoid

42 = 23 + b₂

19 = b₂

b₂ = 19 mm

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Check It Out! Example 2

Find the area of the triangle.

Find b.

A = 96 m²

Substitute 16 for b and 12 for h.

Area of a triangle

Simplify.

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Example 3A: Finding Measurements of Rhombuses and Kites

Find d₂ of a kite in which d₁ = 14 in. and

A = 238 in².

Area of a kite

Substitute 238 for A and 14 for d₁.

Solve for d₂.

Sym. Prop. of =

34 = d₂

d₂ = 34

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Example 3B: Finding Measurements of Rhombuses and Kites

Find the area of a rhombus.

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Substitute (8x+7) for d₁ and (14x-6) for d₂.

Multiply the binomials (FOIL).

Distrib. Prop.

Area of a rhombus

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Example 3C: Finding Measurements of Rhombuses and Kites

Find the area of the kite

Step 1 The diagonals d₁ and d₂ form four right triangles. Use the Pythagorean Theorem to find x and y.

28² + y² = 35²

y² = 441

y = 21

21² + x² = 29²

x² = 400

x = 20

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Step 2 Use d₁ and d₂ to find the area.

d₁ is equal to x + 28, which is 48.

Half of d₂ is equal to 21, so d₂ is equal to 42.

A = 1008 in²

Area of kite

Substitute 48 for d₁ and 42 for d₂.

Simplify.

Example 3C Continued

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Check It Out! Example 3

Find d₂ of a rhombus in which d₁ = 3x m and A = 12xy m².

d₂ = 8y m

Formula for area of a rhombus

Substitute.

Simplify.

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals

Lesson Quiz: Part I

Find each measurement.

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1. the height of the parallelogram, in which

A = 182x² mm²

h = 9.1x mm

2. the perimeter of a rectangle in which h = 8 in. and A = 28x in²

P = (16 + 7x) in.

Holt Geometry

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Developing Formulas for

Triangles and Quadrilaterals

Lesson Quiz: Part II

3. the area of the trapezoid

A = 16.8x ft²

4. the base of a triangle in which

h = 8 cm and A = (12x + 8) cm²

b = (3x + 2) cm

5. the area of the rhombus

A = 1080 m²

Holt Geometry

9-1

Developing Formulas for

Triangles and Quadrilaterals