AREAS

Introduction

In geometry, area refers to the measure of the 2D space enclosed within a boundary or a planar figure. Understanding the areas of polygons, specifically parallelograms and triangles, is a fundamental concept that relies heavily on their base and height, as well as their positioning between parallel lines.

graph TD; Area-->Polygons; Polygons-->Parallelograms; Polygons-->Triangles; Parallelograms-->Formula1["A = base * height"]; Triangles-->Formula2["A = 1/2 * base * height"];

1. Properties of Parallelograms (relating to area)

A parallelogram is a quadrilateral with two pairs of parallel opposite sides.
* Base and Height: Any side of a parallelogram can be considered its base. The perpendicular distance between that base and the opposite parallel side is the height (or altitude).
* Area Formula: $\text{Area} = \text{Base} \times \text{Height}$ ($A = b \times h$).
* A diagonal of a parallelogram divides it into two congruent triangles, which means they have equal areas.

Numerical Example:
Find the area of a parallelogram whose base is $8 \text{ cm}$ and corresponding altitude (height) is $5 \text{ cm}$.
$\text{Area} = b \times h = 8 \text{ cm} \times 5 \text{ cm} = 40 \text{ cm}^2$.

MCQs - Properties of Parallelograms

The area of a parallelogram is calculated by:
a) $\frac{1}{2} \times \text{base} \times \text{height}$
b) $\text{base} \times \text{height}$
c) $2 \times (\text{base} + \text{height})$
d) $\text{base}^2$
Answer: b
A diagonal of a parallelogram divides it into two triangles of:
a) Unequal areas
b) Equal perimeters but unequal areas
c) Equal areas
d) Areas in a 2:1 ratio
Answer: c
If the area of a parallelogram is $50 \text{ cm}^2$ and its base is $10 \text{ cm}$, what is its height?
a) $500 \text{ cm}$
b) $5 \text{ cm}$
c) $10 \text{ cm}$
d) $25 \text{ cm}$
Answer: b
Two parallelograms have the same base and the same height. What can be said about their areas?
a) They are different
b) They are equal
c) One is twice the other
d) Cannot be determined
Answer: b
The altitude of a parallelogram is defined as the:
a) Length of the longest side
b) Distance between two adjacent sides
c) Perpendicular distance between the base and the opposite parallel side
d) Length of the diagonal
Answer: c
What happens to the area of a parallelogram if its base is doubled while the height remains constant?
a) It is halved
b) It is doubled
c) It remains the same
d) It increases by a factor of 4
Answer: b
Which of the following is true for the diagonals of a parallelogram?
a) They divide the parallelogram into four triangles of equal area
b) They are always equal in length
c) They intersect at right angles
d) They are parallel to the sides
Answer: a
If the base of a parallelogram is $15 \text{ cm}$ and its area is $120 \text{ cm}^2$, what is its corresponding height?
a) $8 \text{ cm}$
b) $10 \text{ cm}$
c) $12 \text{ cm}$
d) $15 \text{ cm}$
Answer: a
A square and a parallelogram have the same area. If the square has a side length of $6 \text{ cm}$, and the base of the parallelogram is $9 \text{ cm}$, what is the height of the parallelogram?
a) $2 \text{ cm}$
b) $3 \text{ cm}$
c) $4 \text{ cm}$
d) $6 \text{ cm}$
Answer: c
The distance between the opposite parallel sides of a parallelogram is also known as its:
a) Diagonal
b) Base
c) Median
d) Altitude
Answer: d

2. Same base and between same parallels

Two figures are said to be on the same base and between the same parallels if they have a common base (a common side) and the vertices (or vertex) opposite to the common base lie on a line parallel to the base.
* Parallelograms on the same base and between the same parallels are equal in area.
* This happens because they share the exact same base ($b$) and the exact same perpendicular height ($h$), since the distance between parallel lines is constant.

Practical Example:
Imagine a rectangle $ABCD$ with base $AB = 10 \text{ cm}$ and height $5 \text{ cm}$. Area = $50 \text{ cm}^2$. Now, imagine pulling the top vertices $C$ and $D$ to the right to form a slanted parallelogram $ABEF$, keeping them on the same parallel line. The base is still $10 \text{ cm}$ and the height is still $5 \text{ cm}$. The area of $ABEF$ is still $50 \text{ cm}^2$.

MCQs - Same base and between same parallels

Two parallelograms are on the same base and between the same parallels. The ratio of their areas is:
a) 1:2
b) 2:1
c) 1:1
d) 3:1
Answer: c
For two figures to be between the same parallels, their heights must be:
a) Proportional to their bases
b) Unequal
c) Equal
d) Zero
Answer: c
Parallelogram $ABCD$ and rectangle $ABEF$ are on the same base $AB$ and between the same parallels. If the area of $ABCD$ is $40 \text{ cm}^2$, what is the area of $ABEF$?
a) $20 \text{ cm}^2$
b) $40 \text{ cm}^2$
c) $80 \text{ cm}^2$
d) $10 \text{ cm}^2$
Answer: b
Figures sharing a base and lying between identical parallel lines implies that the distance between the parallel lines acts as their:
a) Common median
b) Common diagonal
c) Common altitude (height)
d) Common perimeter
Answer: c
If a triangle and a parallelogram are on the same base and between the same parallels, what is the relationship between their areas?
a) They are equal
b) The parallelogram's area is half of the triangle's area
c) The triangle's area is half of the parallelogram's area
d) The triangle's area is double the parallelogram's area
Answer: c
If two triangles share a base and have their third vertices on a line parallel to the base, their areas are:
a) In a ratio of 1:2
b) Unequal
c) Equal
d) Dependent on the lengths of their other sides
Answer: c
Two parallelograms, P1 and P2, are on equal bases and between the same parallels. The area of P1 is $45 \text{ cm}^2$. The area of P2 is:
a) $90 \text{ cm}^2$
b) $45 \text{ cm}^2$
c) $22.5 \text{ cm}^2$
d) Cannot be determined
Answer: b
A parallelogram and a square lie on the same base and between the same parallels. If the side of the square is $5 \text{ cm}$, what is the area of the parallelogram?
a) $10 \text{ cm}^2$
b) $20 \text{ cm}^2$
c) $25 \text{ cm}^2$
d) $50 \text{ cm}^2$
Answer: c
The locus of the third vertex of a triangle, given a fixed base and a constant area, is:
a) A circle
b) A line perpendicular to the base
c) A line parallel to the base
d) A point
Answer: c
If the area of a parallelogram on a given base is $80 \text{ sq units}$, the area of any triangle on the same base and between the same parallels is:
a) $40 \text{ sq units}$
b) $80 \text{ sq units}$
c) $160 \text{ sq units}$
d) $20 \text{ sq units}$
Answer: a

3. Theorems on areas

Key theorems regarding the areas of triangles and parallelograms:
* Theorem 1: Two triangles on the same base (or equal bases) and between the same parallels are equal in area.
* Reason: Area of triangle = $\frac{1}{2} \times b \times h$. If they share the base and are between the same parallels (meaning same $h$), their areas must be equal.
* Theorem 2: If a triangle and a parallelogram are on the same base and between the same parallels, the area of the triangle is equal to half the area of the parallelogram.
* Theorem 3: Two triangles having the same base (or equal bases) and equal areas lie between the same parallels. (Converse of Theorem 1).

Numerical Example:
A parallelogram and a triangle share a base of $12 \text{ cm}$ and are situated between the same parallel lines. The height between the parallels is $6 \text{ cm}$.
Area of Parallelogram = $12 \times 6 = 72 \text{ cm}^2$.
Area of Triangle = $\frac{1}{2} \times 12 \times 6 = 36 \text{ cm}^2$. (Notice it is exactly half!)

MCQs - Theorems on areas

If $\triangle ABC$ and $\triangle DBC$ are on the same base $BC$ and between the same parallels, then:
a) $\text{Area}(ABC) = 2 \times \text{Area}(DBC)$
b) $\text{Area}(ABC) = \frac{1}{2} \text{Area}(DBC)$
c) $\text{Area}(ABC) = \text{Area}(DBC)$
d) $\text{Area}(ABC) = \text{Area}(DBC)^2$
Answer: c
A triangle and a parallelogram are on the same base and between the same parallels. If the area of the triangle is $15 \text{ cm}^2$, the area of the parallelogram is:
a) $15 \text{ cm}^2$
b) $7.5 \text{ cm}^2$
c) $30 \text{ cm}^2$
d) $45 \text{ cm}^2$
Answer: c
Two triangles have equal areas and equal bases. This means they must:
a) Be congruent
b) Lie between the same parallel lines
c) Be equilateral triangles
d) Have equal perimeters
Answer: b
The median of a triangle divides it into two triangles of:
a) Equal perimeters
b) Equal areas
c) Unequal areas
d) Congruent shapes always
Answer: b
A rectangle and a rhombus are on the same base and between the same parallels. The ratio of their areas is:
a) 1:2
b) 2:1
c) 1:1
d) Cannot be determined
Answer: c
A parallelogram $PQRS$ and a triangle $PQT$ share the base $PQ$ and lie between parallels $PQ$ and $SR$. If the area of $PQT$ is $24 \text{ cm}^2$, the area of $PQRS$ is:
a) $12 \text{ cm}^2$
b) $24 \text{ cm}^2$
c) $36 \text{ cm}^2$
d) $48 \text{ cm}^2$
Answer: d
Two triangles have equal areas. If one triangle has a base of $10 \text{ cm}$ and an altitude of $6 \text{ cm}$, what is the altitude of the other triangle if its base is $15 \text{ cm}$?
a) $4 \text{ cm}$
b) $5 \text{ cm}$
c) $6 \text{ cm}$
d) $8 \text{ cm}$
Answer: a
The diagonals of a trapezium divide it into four triangles. If the parallel sides are unequal, which pairs of triangles have equal areas?
a) The two triangles adjacent to the parallel sides
b) The two triangles adjacent to the non-parallel sides
c) All four triangles
d) None of the triangles
Answer: b
Let $AD$ be a median of $\triangle ABC$. If the area of $\triangle ABD$ is $x$, the area of $\triangle ABC$ is:
a) $x/2$
b) $x$
c) $2x$
d) $3x$
Answer: c
A rhombus and a rectangle have the same base and the same area. What can be said about their perimeters?
a) The perimeter of the rhombus is greater
b) The perimeter of the rectangle is greater
c) Their perimeters are equal
d) The relationship cannot be determined
Answer: a

4. Solving problems and riders

"Riders" in geometry are essentially practice problems or theorems that require you to apply the main theorems to prove a new property or find a specific value.
* Strategy for Riders:
1. Look for figures sharing a common base.
2. Check if the opposite vertices lie on a line parallel to the base.
3. Use the property that a median divides a triangle into two triangles of equal areas.
4. Use algebraic addition or subtraction of areas (e.g., if $\text{Area}(ABC) = \text{Area}(DEF)$, then $\text{Area}(ABC) - \text{Area}(XYZ) = \text{Area}(DEF) - \text{Area}(XYZ)$).

Numerical Example / Practical Problem:
In $\triangle ABC$, $D$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $50 \text{ cm}^2$, what is the area of $\triangle ABD$?
Solution: Since $D$ is the midpoint of $BC$, $AD$ is a median. A median divides a triangle into two triangles of equal area.
Therefore, $\text{Area}(\triangle ABD) = \frac{1}{2} \times \text{Area}(\triangle ABC) = \frac{1}{2} \times 50 = 25 \text{ cm}^2$.

MCQs - Solving problems and riders

A median of a triangle divides it into two triangles that always have equal:
a) Angles
b) Areas
c) Perimeters
d) Side lengths
Answer: b
In parallelogram $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Which of the following is true?
a) $\text{Area}(AOB) = \frac{1}{4} \text{Area}(ABCD)$
b) $\text{Area}(AOB) = \frac{1}{2} \text{Area}(ABCD)$
c) $\text{Area}(AOB) = \text{Area}(ABC)$
d) Diagonals do not divide the area equally
Answer: a
If $E$ is any point on median $AD$ of $\triangle ABC$, then:
a) $\text{Area}(ABE) > \text{Area}(ACE)$
b) $\text{Area}(ABE) < \text{Area}(ACE)$
c) $\text{Area}(ABE) = \text{Area}(ACE)$
d) $\text{Area}(ABE) = 2 \times \text{Area}(ACE)$
Answer: c
You are given a square and a parallelogram on the same base and between the same parallels. Which figure has a larger perimeter?
a) The square
b) The parallelogram
c) They have equal perimeters
d) Cannot be determined
Answer: b (The slanted sides of the parallelogram will be longer than the straight altitude/side of the square).
If the area of a parallelogram is $60 \text{ sq units}$, and a triangle is formed using the same base and its third vertex is on the opposite parallel side, the area of the remaining part of the parallelogram is:
a) $15 \text{ sq units}$
b) $20 \text{ sq units}$
c) $30 \text{ sq units}$
d) $60 \text{ sq units}$
Answer: c (The triangle takes up half (30), so the remaining part is the other half (30)).
In a trapezium $ABCD$ where $AB \parallel CD$, the diagonals intersect at $O$. The areas of $\triangle AOD$ and $\triangle BOC$ are always:
a) Unequal
b) Equal
c) In a 2:1 ratio
d) Independent of each other
Answer: b
The midpoints of the sides of a quadrilateral are joined in order. The area of the resulting parallelogram is:
a) Half the area of the original quadrilateral
b) Equal to the area of the original quadrilateral
c) Double the area of the original quadrilateral
d) One-fourth the area of the original quadrilateral
Answer: a
If the medians of a triangle intersect at the centroid $G$, then the three triangles formed by joining $G$ to the vertices have:
a) Equal perimeters
b) Equal areas
c) Different areas
d) A ratio of 1:2:3 in areas
Answer: b
In a parallelogram $ABCD$, $P$ is a point on $CD$. The sum of the areas of $\triangle APD$ and $\triangle BPC$ is equal to:
a) The area of $\triangle APB$
b) Half the area of $\triangle APB$
c) Double the area of $\triangle APB$
d) The area of the parallelogram
Answer: a
If the area of a rhombus is $96 \text{ cm}^2$ and one of its diagonals is $16 \text{ cm}$, the length of the other diagonal is:
a) $6 \text{ cm}$
b) $8 \text{ cm}$
c) $12 \text{ cm}$
d) $14 \text{ cm}$
Answer: c

Fun Facts about Areas in Geometry!

The Origin of Geometry: The word Geometry comes from Greek words 'geo' (earth) and 'metron' (measurement). Ancient Egyptians developed area formulas primarily to redraw farm boundaries after the Nile river flooded every year!
Shearing: The concept that parallelograms on the same base and between the same parallels have the same area is related to a physical transformation called "shearing" (like pushing a deck of cards sideways). The area doesn't change!
Cavalieri's Principle: An extension of these area theorems to 3D space is called Cavalieri's Principle. It states that if two solids have the same height and the same cross-sectional area at every level, they have the same volume.
Infinite Shapes, Same Area: Between two parallel lines, you can draw an infinite number of different-looking triangles on a single base, and every single one of them will have the exact same area.
Maximum Area: If you have a given perimeter and want to enclose the maximum possible area using a quadrilateral, you should always form a square!