Ratio, proportion, and scale. ☰

Ratio, proportion, and scale are important concepts in mathematics and are frequently used in many different fields, including engineering, science, and finance. Here is a brief explanation of each concept:

Ratio: A ratio is a way of comparing two or more quantities. It is expressed as the quotient of two numbers and can be written in different ways, such as with a colon \( (3 \div 5) \), with a fraction \( \frac{3}{5} \) or as a decimal \((0.6)\). Ratios can be used to describe a wide range of relationships, such as the ratio of boys to girls in a class, or the ratio of the area of a square to the area of a circle.

Proportion: A proportion is a statement that two ratios are equal. In other words, if \( \frac{a}{b}=\frac{c}{d}\) , then we say that \(a\), \(b\), \(c\) and \(d\) are in proportion.
In mathematics, a proportion is a statement that two ratios are equal. A proportion can be written as \( \frac{a}{b}=\frac{c}{d}\), or as \( a \div b = c \div d \).

The properties of proportion are:

• Multiplying or dividing both terms of a ratio by the same non-zero number does not change the ratio or the proportionality relationship.
  For example, if \( \frac{a}{b}=\frac{c}{d}\) , then \( \frac{2a}{2b}=\frac{2c}{2d}\) and \( \frac{a}{2b}=\frac{c}{2d}\)
• If \( \frac{a}{b}=\frac{c}{d}\) , then \( \frac{b}{a}=\frac{d}{c}\). This is called the inverse property of proportion. It means that if we interchange the numerator and denominator of each ratio, we still have a valid proportion.
• If \(\frac{a}{b}=\frac{c}{d}\) and \(\frac{c}{d}=\frac{e}{f}\) , then \(\frac{a}{b}=\frac{e}{f}\). This is called the transitive property of proportion. It means that if two ratios are equal to a third ratio, then they are also equal to each other.
• If \( \frac{a}{b}=\frac{c}{d}\) , then \(\frac{a+b}{b}=\frac{c+d}{d}\). This is called the addition property of proportion. It means that if we add the two terms of each ratio, we still have a valid proportion.
• If \( \frac{a}{b}=\frac{c}{d}\) , then \(\frac{a-b}{b}=\frac{c-d}{d}\). This is called the subtraction property of proportion. It means that if we subtract the second term from the first term of each ratio, we still have a valid proportion.

Scale: Scale is a measure of the size or magnitude of something relative to a standard or reference point. In mathematics, scale is often used to represent distances or measurements in a diagram or map. For example, a map might use a scale of \( 1:1000 \), which means that 1 unit on the map represents 1000 units in the real world. Scales can also be used to represent the size of objects, such as in architectural drawings or engineering diagrams.

Proportional parts ☰

The topic of proportional parts is a fundamental concept in geometry that involves the relationship between the lengths of different line segments. When two or more line segments are proportional, it means that their lengths are related in a fixed way.
The theorem of Thales is a specific example of a proportional parts theorem that applies to triangles. It states that if a line is drawn parallel to one side of a triangle, then it divides the other two sides of the triangle into proportional parts.
In other words, if a line is drawn parallel to one side of a triangle, then the two segments it creates on the other two sides of the triangle are proportional to the length of the original side.
This can be written mathematically as: \( \frac{AB}{AC} = \frac{DE}{DF} \), where \(AB\) and \(AC\) are two sides of a triangle with \(D\) as a point on \(AC\) and \(E\) and \(F\) on \(AB\) and \(BC\) respectively, and \(DE\) and \(DF\) are the two segments created by the parallel line.
This theorem is named after Thales of Miletus, an ancient Greek philosopher and mathematician who is credited with discovering it. The theorem of Thales is used in many areas of mathematics and science, including trigonometry, physics, and engineering.
The theorem of Thales is a powerful tool for solving problems involving triangles and their side lengths. It allows us to find missing side lengths or to prove relationships between different parts of a triangle. It is also a key concept in the study of similar triangles, which are triangles that have the same shape but different sizes.

Medians and bisectors of a triangle.
In geometry, medians and bisectors are two important concepts associated with triangles.

A median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. In other words, each vertex of a triangle has a corresponding median, which divides the opposite side into two equal segments. The point where the three medians of a triangle intersect is called the centroid. The centroid is often referred to as the center of mass of the triangle, as it is the point where the triangle would balance perfectly if it were cut out of a uniform material.

The medians of a triangle have several important properties, including:
1. Each median divides the triangle into two smaller triangles of equal area.
2. The length of each median is equal to one-half of the length of the corresponding side of the triangle.
3. The medians of a triangle are concurrent, meaning that they all intersect at the same point (the centroid).

A bisector of a triangle is a line or line segment that divides one of the angles of the triangle into two congruent angles. In other words, each angle of a triangle has a corresponding angle bisector. The three angle bisectors of a triangle are concurrent, meaning that they all intersect at the same point (called the incenter). The incenter is the center of the inscribed circle, which is a circle that is tangent to all three sides of the triangle.

The angle bisectors of a triangle have several important properties, including:
1. The incenter is equidistant from the three sides of the triangle.
2. The angle bisectors of a triangle divide the opposite side into two segments that are proportional to the adjacent sides of the triangle.
3. The length of each angle bisector can be calculated using the formula:
\( l_A = \frac{2ab cos \frac{C}{2}}{a+b} \) where \(a\) and \(b\) are the lengths of the adjacent sides of the angle being bisected, and \(C\) is the measure of the angle.

Similar quadrilaterals, similar triangles. ☰

In geometry, two figures are said to be similar if they have the same shape, but not necessarily the same size. When two figures are similar, their corresponding angles are congruent and their corresponding sides are proportional.
Similarity is an important concept in geometry, as it allows us to compare and analyze figures that may be different sizes or orientations. Two common types of similar figures are similar quadrilaterals and similar triangles.

Similar Quadrilaterals:
Two quadrilaterals are similar if their corresponding angles are congruent and their corresponding sides are proportional. This means that we can create one quadrilateral by enlarging or shrinking the other quadrilateral while keeping the same shape.
One important property of similar quadrilaterals is that the ratios of their corresponding sides are equal. This means that if we know the ratio of the lengths of one set of corresponding sides, we can find the ratio of the lengths of all the other corresponding sides.

Similar Triangles:
Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional. This means that we can create one triangle by enlarging or shrinking the other triangle while keeping the same shape.
One important property of similar triangles is that the ratios of their corresponding sides are equal. This property is known as the "Triangle Proportionality Theorem" or the "Side-Splitter Theorem". This means that if we know the ratio of the lengths of one set of corresponding sides, we can find the ratio of the lengths of all the other corresponding sides.
Similar triangles are used in many areas of mathematics and science, including trigonometry, physics, and engineering. They are also used in art and design, as similar triangles can be used to create perspective and depth in drawings and paintings.

Similarity signs of triangles and similarity of right triangles. ☰

In geometry, there are two symbols used to indicate similarity between two triangles:
The symbol \( \sim \) : Two triangles are similar if they have the same shape, but not necessarily the same size. We use the symbol \( \sim \) to indicate that two triangles are similar. For example, \( \Delta ABC \sim \Delta XYZ \) indicates that triangles \(ABC\) and \(XYZ\) are similar.
The symbol \( \cong \): Two triangles are congruent if they have the same shape and size. We use the symbol \( \cong \) to indicate that two triangles are congruent. For example, \( \Delta ABC \cong \Delta XYZ \) indicates that triangles \(ABC\) and \(XYZ\) are congruent.
When it comes to right triangles, there are some additional properties to consider. If two right triangles have one acute angle that is congruent and the length of the hypotenuse of one triangle is proportional to the length of the hypotenuse of the other triangle, then the two right triangles are similar. This property is known as the "Hypotenuse-Leg" or "HL" similarity criterion.

Another important property of similar right triangles is the ratio of the lengths of their corresponding sides. This ratio is known as the "trigonometric ratios" and is defined as follows:

• Sine: The sine of an acute angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine: The cosine of an acute angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent: The tangent of an acute angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

These trigonometric ratios are used extensively in trigonometry, which is the branch of mathematics that deals with the relationships between angles and sides of triangles.

Area of similar figures ☰

Similar figures are two or more figures that have the same shape but may be different in size. For instance, two rectangles can be similar even though one is larger than the other. The area of similar figures can be determined using the following theorems and formulas:

• Theorem of Proportional Sides: If two figures are similar, then the ratio of their corresponding sides is equal. In other words, if two rectangles are similar, then the ratio of their lengths and widths will be the same.
• Theorem of Proportional Perimeters: If two figures are similar, then the ratio of their perimeters is equal to the ratio of their corresponding sides. In other words, if two rectangles are similar, then the ratio of their perimeters will be the same as the ratio of their lengths and widths.
• Theorem of Proportional Areas: If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding sides. In other words, if two rectangles are similar, then the ratio of their areas will be the same as the square of the ratio of their lengths and widths.

With these theorems, we can use the following formulas to find the area of similar figures:

• If two rectangles are similar, then the ratio of their areas is equal to the square of the ratio of their lengths and widths:
  
  \( \frac{A_1}{A_2}=(\frac{l_1}{l_2})^2 = (\frac{w_1}{w_2})^2\) where \(A_1\) and \(A_2\) are the areas of the two rectangles, \(l_1\) and \(l_2\) are their lengths, and \(w_1\) and \(w_2\) are their widths.
• If two triangles are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding sides:
  
  \( \frac{A_1}{A_2}=(\frac{S_1}{S_2})^2\) where \(A_1\) and \(A_2\) are the areas of the two triangles, and \(s_1\) and \(s_2\) are the lengths of their corresponding sides.
• If two circles are similar, then the ratio of their areas is equal to the square of the ratio of their radii:
  
  \( \frac{A_1}{A_2}=(\frac{r_1}{r_2})^2\) where \(A_1\) and \(A_2\) are the areas of the two circles, and \(r_1\) and \(r_2\) are their radii.

In summary, to find the area of similar figures, we use the theorem of proportional areas along with the appropriate formula for the type of figures we are working with.