MATHS –CLASS IX Topic : AREA OF Parallelogram

Area Of Parallelograms Definition: A parallelogram is a quadrilateral with opposite sides parallel.

Review of Characteristics of Parallelograms Opposite sides are parallel. Consecutive angles are supplementary. (SSI angles) Opposite sides are congruent. Opposite angles are congruent.

Review of Characteristics of Parallelograms Opposite sides are parallel. Consecutive angles are supplementary. (SSI angels) Opposite sides are congruent. Opposite angles are congruent. Diagonals bisect each other.

Areas of Parallelograms Find the area by counting squares. 8 Full blocks = 4 Half blocks = Total Area = 10 su.

Areas of Parallelograms Find the area by counting squares. Full blocks = 12 Half blocks = 6 Total Area = 15 su.

Areas of Parallelograms Find the area by counting squares. 20 Full blocks = 8 Half blocks = Total Area = 24 su.

There is!!! Don’t you agree? ? ? There must be ? ? an easier way! ? ? It is tough to compute all the partial blocks. ? ? There must be an easier way! ? ? ? ? There is!!! ? ? I like rectangles much better! They are real easy! Don’t you agree?

Let’s cut off a corner and start to make a rectangle. Doesn’t a parallelogram look like a rectangle with it’s side kicked in? Let’s cut off a corner and start to make a rectangle.

Wow! We have part of a rectangle. Now watch what else we have.

Let’s move the triangle to the other side. The two triangles are congruent. Let’s move the triangle to the other side.

What has happened? We have now created a rectangle with the same area as the parallelogram.

We have now created a rectangle with the same area as the parallelogram. Therefore, the formula for the area of a parallelogram is the same as that of a rectangle.

Parallelogram h b

Parallelogram b2 h2 Note that there is another base and another height. Sometimes you must use the other height.

Triangle Now, we can use the parallelogram formula to derive No, we are not going to add up squares again. Now, we can use the parallelogram formula to derive the area of a triangle.

Triangle Let’s construct a line through the vertex parallel to the base.

Triangle We have just created a parallelogram. Let’s construct another line through the right vertex parallel to another side. We have just created a parallelogram.

Triangle SAS, SSS, ASA, or AAS We can do this for any triangle. Note that both the triangles are congruent by… SAS, SSS, ASA, or AAS

SAS Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. Opposite sides of a parallelogram are congruent. SAS

SSS Opposite sides of a parallelogram are congruent. Reflexive property. SSS

ASA AIA: Alternate Interior Angles are congruent. Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. ASA

AAS AIA: Alternate Interior Angles are congruent. Opposite angles of a parallelogram are congruent. Opposite sides of a parallelogram are congruent. AAS

Triangle All triangles are just Half a parallelogram. Therefore…

Triangle Note where the height is located. It is the height of both the parallelogram and triangle. h

Sample Problems A= 121.6875 A= 121.7 Find the area. 8.25 14.75 14.75

Finding Areas of Parallelogram 12 16 20 12 16

Find the area: fractions

Find the area: fractions Convert to decimals.

Find the area: mixed modes Convert to decimals.

Find the area: mixed modes Convert to decimals. A = 59.754 A = 59.8

Find Area of Triangle 5 3 4 3 4

Find the base associated with the corresponding height. Triangle ABC has three altitudes or heights. They are … Each side is a corresponding base. Find the base associated with the corresponding height.

h h If the area = 96 and …. Find the values of AD and FC. This is a backwards problem. You always start with the formula. h

h h If the area = 96 and …. Find the values of AD and FC. This is a backwards problem. You always start with the formula. h

Backward Problems 2 2 h 25 h 125 25 If the area is 125 sf, and the base is 25, find the height. h 2 2 25 Divide by 25 h 125 25

182 h 30 30 h If the area is 182 sf, and the base is 30, find the height. h 30 182 30 h

Compound Complex Multiple Stage Problems With these problems, you… 1 Plan solution with equations or written strategy as if you have all the information needed. 2. Write the equation, leaving empty parentheses to insert the needed values. 3. Go find the needed values in sidebar stages, substituting back into the original strategy or equation. 4. When all values are found, complete the original strategy or equation.

Find Area of Triangle 13 5 12 12 5 5, 12, 13 triangle

Find Area of Triangle 8 8 600 8 4 8

Find Area of Triangle 8 15 15 600 30-60-90 triangle

Find Area of Triangle 11 5 b 5 b 11 5

Find Area of Parallelogram 8 h 16 16 Need to find value of height.

Find Area of Triangle 8 15 15 8 45-45-90 triangle

Find the area of pentagon. Add line and label figure. 600 10 5 10 10 10

They all have the same area. Why? Which triangle has the largest area? They all have the same area. Why?

They each have the same base: AB They each have the same height.

h h is 13 cm. M is midpoint of Find the area of triangles ABM and ACM. h is 13 for both triangles.

M is midpoint of h is 13 cm. h

Summary 1. The area of a parallelogram is… 2. The area of a triangle is half the area of parallelogram. 3. A triangle has three heights or altitudes. 4. A triangle has three bases (sides) to correspond with each height.