Top Ten Properties of a Rectangle

Top Ten Properties of a Rectangle

Rectangle

1. A rectangle is a quadrilateral so it has four sides.

2. A rectangle is a parallelogram

3. The perimeter of a rectangle equals 2(base X height)

4. The area of a rectangle equals base X height

5. Opposite sides of a rectangle are congruent

6. A rectangle has four right angles

7. The diagonal of a rectangle is the hypotenuse of a right triangle

8. Opposite sides of a rectangle are parallel

9. The diagonals of a rectangle do not intersect in a right angle.

10. The diagonals of a rectangle create opposite central angles that are congruent.

Complementary and Supplementary Angles

 

What are complementary and supplementary angles?

Complementary angles are two angles that add to ninety degrees.

Supplementary angles are two angles that add to one hundred and eighty degrees.

Sample problem one.

30 is the complement of what angle?

Step 1 30 + X = 90◦

Step 2 Subtract 30 from each side x= 60◦

Sample problem 2 What is the angle measure of X ?

 

 

Step 1 In a right triangle the two angles are complementary.

Step 2 90 – 68= 22◦

Sample problem 3

105◦ is the supplement of what angle?

Step 1 x + 105 = 180

Step 2 subtract 105 from each side x = 75◦

Sample problem 4

What is the angle measure of ∠x ?

 

 

Step 1 A straight line measures 180◦

Step 2 180 -65 = 115◦ is the measure of ∠x

Finding the surface area of a Cone

How to find the surface area of a Cone

The video works the problem Surface Area of a Cone

 

In order to find the surface area of a cone you add the lateral area plus the base area

The lateral area of a cone equals 1/2 dπ*s  

D = diameter

S = slant height

Base Area = πr^2

Find the surface area of a cone with a diameter of 6 units and a height of 4 units.

Step 1. The radius = ½ diameter = 3 units

Step 2 Find the slant height by using the Pythagorean Theorem a^2+b^2=c^2

Use the height and radius to find the slant height  3^2+4^2=c^2 which equals 9 +16 = 25

√25=5 equals your slant height 

Step 3 Now use 1/2 dπ*s  

1/2 6π*5=3π*5=15π which will equals the lateral area 

Step 4. Find the base area  by using πr^2

π3^2 = π9

Step 5 Add the lateral and base area in order to get surface area

15π+9 π=24π units^2

Check out the Video if confused Surface Area of a Cone

How to find the perimeter of a parallelogram

 

Perimeter of a Parallelogram

Watch the video to see each problem worked out

Perimeter of a Parallelogram

In order to find the perimeter of a parallelogram you can use two methods.

Method 1 Perimeter of parallelogram equals, add all four sides

 

Method 2 Perimeter of parallelogram equals 2 l +2w l = length w = width

 

Problem 1 Find the perimeter of a parallelogram with a side of 3 units and 10 units.

 

 

 

Step 1. The opposite sides of a parallelogram are congruent. Therefore, if you know the length of one side you know the length of the other.

 

Step 2. Add all four sides 3+3+10+10=26 units or (2*3) + (2*10) = 6+20=26 units

 

Problem 2 Find the perimeter of a parallelogram with a side of 8 units and a height of 6 units and an angle measure of 60 degrees.

 

 

 

Step 1 Use the altitude to find the length of the missing side. 

 

Step 2 The altitude creates a 30-60-90 triangle. If you find the hypothesis of the 30-60-90 Triangle this will be the length of the missing side.

 

Step 3 The height becomes the long leg which equals x√(3 ) x = the length of the short leg

  

Since you know the length of the long leg you can use 6=x√3 

  

Step 3A 6/√3 = √3/√3  

  

Step 3B (6√3)/2 =2√3  

 

Step 4 Now that I know x I can use 2x to find the hypotenuse

  2*2√(3 ) =4√3  

 

Step 5. Now use the perimeter formula for a parallelogram 2l + 2w

   

2*8 + 2*4√3

  

16 + 8√3 this is your final answer because you can add a constant and a radical

Perimeter of a rhombus

Finding the perimeter of a rhombus

The video works out each problem

Perimeter of a rhombus

Let’s look at finding the perimeter of a rhombus. The perimeter is the distance around the outside of an object. A rhombus is similar to a square with a couple of unique features. They are both quadrilaterals but a rhombus doesn’t have right angles at the corners. Like a square a rhombus has congruent sides, so the perimeter formula is the same as a square, which is 4 times one side length.

Perimeter = 4s s=side

Problem 1. What is the perimeter of a rhombus with a side of 5 units?

 

 The video works the problem

Perimeter of a rhombus

Step 1. Use 4s

Step 2 4*5= 20 units Perimeter is linear, therefore it is not squared

Let’s next look at a problem a little more involved.

Find the perimeter of a rhombus with a diagonal of 10 units and measure of angle ABC equals 120 degrees.

 

 

Step 1. The diagonals  of a rhombus are  perpendicular to each other so they form right angles. These right angles  create a right triangle.

Step 2. The diagonal also bisects each other and divide each other in half. The 10 unit diagonal is divided into two 5 unit lines Combine step one and two  and you have a 30-60-90 right triangle  with a short leg of 5 units .

 

 

 

Step 3. In order to find the side of the rhombus I will find the hypotenuse of the right triangle. The formula for the hypotenuse of a right triangle equals 2x

Step 4. We know x = 5 units from the diagonal being bisected by the other diagonal so it is 5 units.

So the hypotenuse equals 2*5 = 10 units

Step 5. Now use the perimeter formula of 4s which equals 4* 10 = 40 units

Finding the perimeter of a rectangle.

How to find the perimeter of a rectangle.

 The video link below works each problem. Hope it helps in class.

How to find the perimeter of a rectangle

Let’s look at the step by step procedure  for finding the perimeter of a rectangle. First we need to look at some rectangle properties. A rectangle has opposite sides that are equal  and four right angles. Therefore if you know the length of one side then the opposite side will be the same length. The right angles allow the  diagonal to divide the rectangle into a right triangle.

 

Example problem 1. Find the perimeter of a rectangle with one side of 8 units and one side of 12 units.

 

 

 

Step 1. Find the length of the two missing sides, this is easy because opposite sides are congruent so the four sides are 8, 8, 12, 12 units.

 

Step 2. Add the side lengths together. 

8+8+12+12=40 units

 

Step 3. Another method to find the perimeter of a rectangle is to use 2*length + 2* Width

2*8 + 2*12

16 + 24 = 40 units

 

Example problem 2. Find the perimeter if you have a rectangle with a diagonal of 13 units and a side length of 12 units. 

 

 

 

 

Step 1. Because a rectangle has four right angles the diagonal creates a right triangle with the diagonal being the hypotenuse of the right triangle.

 

Step 2. Use the Pythagorean Theorem to find the length of the missing side. Let’s label c=13 (the diagonal) a = 12 (the side length we are given) and solve for b

 

Step 3. Use the Pythagorean Theorem a^2+b^2=c^2 

Step a.12^2+b^2=13^2 

 

Step b.144+b^2=169 (Subtract 144 from each side.

 

Step c. b^2=25  

 

Step d.  b=√(25 ) b=5 units

 

Step 4. Now we know the missing side so I can use the formula for the perimeter of a rectangle

2*length + 2*width = perimeter

2*5 + 2*12 = 10 +24= 34 units

How to find the area of a square.

How to find the area of a square ?

Please watch the video if you need to watch the problems being solved

Area of a square

 

Let’s begin with the formula for area of a square is s^2   s=side

 Example 1 Find the area of a square with a side of 8 units. 

 

 

Step 1 Plug the number into the formula: 8^(2 )=64 units^2

Let’s look at an example using diagonals.

 

 

 

 

 

Example 2 Find the area of a square with a diagonal of 14 units.

 

Step 1. The diagonal bisects the ninety degree angle and creates a 45-45-90 triangle. We can use the rules of a 45-45-90 triangle to find the length of one side.

 

Step 2. The leg lengths of a 45-45-90 triangle equals x and the hypotenuse equals x√2

 

Step 3. In order to get x by itself I will take the hypotenuse length of the 45-45-90 triangle that the diagonal creates 14 and set it equal to x root 2 which looks like

14=x√2 

 

Step 4. Solve for x by dividing by the √2

14/√2=(x√2)/√2 =7√2 = S

 

Step 5.Now we plug s into our area formula for a square s^2

 

Step 6. (7√(2 ) )^2 = 42 x2 =98 units^2

What is so special about a 45-45-90 triangle?

What is so special about a-45-90-45 Triangle?

A 45-45-90 triangle is classified as a special right triangle, but why is it special? Let’s look at the special rules of a 45-45-90 Triangle.

Rule 1. If you know the length of one leg you know the length of the other leg because they are equal.

Rule 2. In order to find the length of the hypotenuse multiply the leg length times the square root of two. This would be written x√2 where x=leg length

Rule 3. In order to find the leg length divide your hypotenuse by the square root of 2.

 

Rule 4 .The area of a 45-45-90 triangle equals 1/2(leg)^2

Let’s look at two examples. 

Example 1 Find the hypotenuse of a 45-45-90 triangle with leg length of 7 units.

 

Step 1. Use x√(2 ) 

Step 2. 7√(2 ) units

Example2. What is the length of one leg of a 45-45-90 triangle if you have a hypotenuse with a length of 12 units?

Step 1. 12/√2  use the formula for leg length hypotenuse divided by square root two

Step 2. 12/√2* √2/√2 

Step 3. (12√2)/2 =6√2

What is a Quadrilateral?

What is a quadrilateral? The key to a quadrilateral is found in its name, “quad" meaning four and “lateral" meaning side. So a quadrilateral is a four sided polygon. Some common examples are kites, parallelograms, trapezoids, rhombuses, rectangles, and squares. 

The following video goes over the properties of a Quadrilateral

Properties of a Quadrilateral

• Four Sided

• Closed figure

• Four vertices (also known as corners)

• Interior angles add up to 360◦

• Straight sides, no curves

  Two diagonals

 

Quadrilaterals can be subdivided into parallelograms and trapezoids. Let’s look at parallelograms. Some common examples of parallelograms are rectangles, rhombuses, and squares. They all have opposite sides that are parallel and congruent, and diagonals that bisect each other.

Trapezoids have exactly one pair of parallel sides. The perimeter of a trapezoid is the sum of four side lengths and the area is one half times base one plus base two times height. The perimeter and area of each quadrilateral varies according to its shape. Here is a link to the perimeter and area formulas for most quadrilaterals.