7 3 Similar Triangles Skills Practice

7.3 Similar Triangles: Skills Practice and Mastery

Similar triangles are a cornerstone of geometry, offering a powerful tool for solving a vast range of problems. Understanding their properties and applying the relevant theorems is crucial for success in geometry and related fields like trigonometry and calculus. This comprehensive guide delves into the concept of similar triangles, providing a thorough skills practice session with various examples and problem-solving strategies. We'll explore the different ways similar triangles can be identified and used to calculate unknown lengths and angles.

Understanding Similar Triangles: Key Concepts

Similar triangles are triangles that have the same shape but not necessarily the same size. This means their corresponding angles are congruent (equal), and their corresponding sides are proportional. This proportionality is key to solving many problems. Let's break down the essential elements:

• Corresponding Angles: If two triangles are similar, their corresponding angles are equal. For example, if triangle ABC is similar to triangle DEF (written as ∆ABC ~ ∆DEF), then ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F.

• Corresponding Sides: The ratio of the lengths of corresponding sides is constant. This constant ratio is known as the scale factor. For ∆ABC ~ ∆DEF, the following ratios hold true: AB/DE = BC/EF = AC/DF.

Identifying Similar Triangles: Essential Theorems

Several theorems help us identify similar triangles:

• AA (Angle-Angle) Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Remember, if two angles are equal, the third angle must also be equal (because the sum of angles in a triangle is always 180°).

• SSS (Side-Side-Side) Similarity Theorem: If the ratios of the corresponding sides of two triangles are equal, then the triangles are similar.

• SAS (Side-Angle-Side) Similarity Theorem: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

Skills Practice: Solving Problems with Similar Triangles

Let's put these concepts into practice with a variety of problems, progressively increasing in complexity.

Problem 1: Basic Proportionality

Two triangles, ∆ABC and ∆DEF, are similar. The lengths of the sides of ∆ABC are AB = 6 cm, BC = 8 cm, and AC = 10 cm. If DE = 3 cm, find the lengths of EF and DF.

Solution:

Since ∆ABC ~ ∆DEF, the ratios of corresponding sides are equal. We have:

AB/DE = BC/EF = AC/DF

Substituting the given values:

6/3 = 8/EF = 10/DF

Solving for EF:

2 = 8/EF => EF = 4 cm

Solving for DF:

2 = 10/DF => DF = 5 cm

Problem 2: Using the AA Similarity Postulate

Two triangles, ∆PQR and ∆STU, have ∠P = ∠S = 50° and ∠Q = ∠T = 70°. Are the triangles similar? Explain your reasoning.

Solution:

Yes, the triangles are similar by the AA Similarity Postulate. Since ∠P = ∠S and ∠Q = ∠T, the third angles must also be equal (∠R = ∠U = 180° - 50° - 70° = 60°). Having two pairs of congruent angles is sufficient to prove similarity.

Problem 3: Application of the SSS Similarity Theorem

Triangle XYZ has sides XY = 9 cm, YZ = 12 cm, and XZ = 15 cm. Triangle RST has sides RS = 6 cm, ST = 8 cm, and RT = 10 cm. Are the triangles similar? If so, what is the scale factor?

Solution:

Let's check the ratios of corresponding sides:

XY/RS = 9/6 = 1.5

YZ/ST = 12/8 = 1.5

XZ/RT = 15/10 = 1.5

Since all the ratios are equal, the triangles are similar by the SSS Similarity Theorem. The scale factor is 1.5; ∆XYZ is larger than ∆RST.

Problem 4: Using Similar Triangles to Find Unknown Lengths

A person who is 1.8 meters tall casts a shadow of 2.4 meters. At the same time, a building casts a shadow of 18 meters. How tall is the building?

Solution:

We can set up similar triangles using the person and the building. Let 'h' be the height of the building.

Person's height / Person's shadow = Building's height / Building's shadow

1.8 / 2.4 = h / 18

Solving for h:

h = (1.8 * 18) / 2.4 = 13.5 meters

The building is 13.5 meters tall.

Problem 5: More Complex Application – Overlapping Triangles

Two triangles overlap. In the larger triangle, two sides have lengths 15 cm and 20 cm, with an included angle of 60°. In the smaller triangle, two sides have lengths 9 cm and 12 cm, with an included angle of 60°. Are these triangles similar?

Solution:

Yes, these triangles are similar by the SAS Similarity Theorem. The ratio of the two known sides in the larger triangle is 20/15 = 4/3. The ratio of the two known sides in the smaller triangle is 12/9 = 4/3. Since these ratios are equal and the included angles (60°) are congruent, the triangles are similar.

Problem 6: Using Similar Triangles in Real-World Scenarios

A surveyor needs to determine the width of a river. They stand at point A, directly across from a tree at point B. They walk 100 meters along the riverbank to point C. They then measure the angle ACB, which is 60°. They also know that angle ABC is 90°. Find the width of the river (AB).

Solution:

This problem utilizes trigonometric functions within the context of similar triangles. While not strictly a similar triangle problem, the principle of proportion is crucial here. The relationship between the angles and sides of right-angled triangles is fundamental to this concept. We can utilize the tangent function:

tan(60°) = AB/AC

AB = AC * tan(60°) = 100 * √3 ≈ 173.2 meters

The width of the river is approximately 173.2 meters.

Problem 7: Advanced Application – Indirect Measurement

A student wants to estimate the height of a tall tree. They measure the length of the tree's shadow (15 meters) and their own shadow (1.5 meters). The student is 1.6 meters tall. Estimate the tree's height.

Solution:

We can use similar triangles to solve this.

Tree's height / Tree's shadow = Student's height / Student's shadow

h / 15 = 1.6 / 1.5

Solving for h:

h = (1.6 * 15) / 1.5 = 16 meters

The estimated height of the tree is 16 meters.

Further Practice and Enrichment

This guide provides a strong foundation in understanding and applying the concepts of similar triangles. To further enhance your skills, consider exploring additional resources, working through more complex problems, and researching real-world applications of similar triangles in fields like architecture, engineering, and cartography. Remember, consistent practice is key to mastering any mathematical concept. The more problems you solve, the more intuitive the application of similar triangles will become. By diligently working through these problems, you’ll develop not only a strong understanding of the theoretical basis of similar triangles but also a practical ability to use them to solve a wide range of problems. Continue exploring the fascinating world of geometry and the powerful tools it offers!