Table of Contents
Area of Two Similar Triangles
The area of two similar triangles is equal to the product of the corresponding sides of the triangles divided by the square of the sine of the angle of similarity.
Properties of Area of Similar Triangles
The properties of the area of similar triangles are that the areas of the triangles are proportional to the squares of the corresponding sides. This means that if two triangles are similar, then the ratio of the areas of the two triangles is the same as the ratio of the corresponding sides. Additionally, the area of a triangle is found by multiplying the base by the height and dividing by two.
Formulas of Area of Similar Triangles
The area of a triangle can be found using the following formula:
A = b x h
Where b is the base of the triangle and h is the height.
The area of a triangle is also equal to one-half the base multiplied by the height.
A = 1/2 b x h
Similar Triangles and Congruent Triangles
A triangle is congruent to another triangle if and only if the three angles of the first triangle are congruent to the three angles of the second triangle.
1. Angle-Angle Similarity or (AAA)
Theorem: If two angles are congruent, then they are also similar.
Proof:
Angles are congruent if and only if they have the same measure. We can see that if two angles are similar, then they must have the same measure because the angles are formed by the same lines and the ratios of the corresponding sides are the same.
\begin{align*}\angle A&=\angle B\\ \frac{AB}{BC}&=\frac{BA}{CA}\\ \angle A&=\angle B\\ \)\end{align*} Angle A=Angle B
\\\frac{AB}{BC}=\frac{BA}{CA}\\\angle A=\angle B\\
Since the angles are congruent and the corresponding ratios of the sides are the same, it follows that the angles are also similar.
2. Side-Side-Side Similarity (SSS)
SSS similarity is the similarity between three objects that are placed in a line next to each other. The objects are usually aligned so that their corresponding sides are parallel to each other.
In the image below, the three squares are similar because their corresponding sides are parallel to each other. The top and bottom squares are also similar because their corresponding sides are perpendicular to each other.
3. Side-Angle-Side Similarity or (SAS)
The Side-Angle-Side Similarity postulate states that two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
