7 3 Practice Similar Triangles

Mastering Similar Triangles: 7 Proven Practice Problems

Understanding similar triangles is crucial for success in geometry and beyond. This concept forms the foundation for many advanced mathematical principles and finds practical applications in fields like architecture, engineering, and surveying. This comprehensive guide will walk you through seven practice problems designed to solidify your understanding of similar triangles, encompassing various difficulty levels and problem-solving strategies. We'll explore the underlying theorems and provide detailed solutions, equipping you with the tools to tackle any similar triangle challenge. Keywords: similar triangles, geometry, congruence, ratios, proportions, solving for unknowns, AAA similarity, SAS similarity, SSS similarity.

Introduction to Similar Triangles

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 – it allows us to set up ratios and solve for unknown side lengths or angles. The most common methods for proving triangle similarity are:

• AAA (Angle-Angle-Angle): If all three angles of one triangle are congruent to the three angles of another triangle, the triangles are similar.
• SAS (Side-Angle-Side): If two sides of one triangle are proportional to two sides of another triangle, and the included angle is congruent, the triangles are similar.
• SSS (Side-Side-Side): If all three sides of one triangle are proportional to the three sides of another triangle, the triangles are similar.

Problem 1: Basic Proportionality

Problem: Triangle ABC is similar to triangle DEF. AB = 6 cm, BC = 8 cm, and DE = 9 cm. Find the length of EF.

Solution: Since triangles ABC and DEF are similar, their corresponding sides are proportional. We can set up a proportion:

AB/DE = BC/EF

6/9 = 8/EF

Cross-multiply: 6EF = 72

Solve for EF: EF = 72/6 = 12 cm

Problem 2: Finding an Unknown Angle

Problem: Triangle PQR is similar to triangle XYZ. ∠P = 50°, ∠Q = 70°, and ∠X = 50°. Find the measure of ∠Z.

Solution: Because triangles PQR and XYZ are similar, their corresponding angles are congruent. Since ∠P = ∠X = 50°, we know that ∠R corresponds to ∠Z. The sum of angles in a triangle is 180°. Therefore:

∠R = 180° - ∠P - ∠Q = 180° - 50° - 70° = 60°

Since ∠R corresponds to ∠Z, ∠Z = 60°.

Problem 3: Applying SAS Similarity

Problem: Two triangles, ABC and DEF, have AB = 4 cm, BC = 6 cm, ∠B = 75°, DE = 6 cm, and EF = 9 cm. ∠E = 75°. Are the triangles similar? Explain.

Solution: We can use the SAS similarity criterion. We have:

AB/DE = 4/6 = 2/3

BC/EF = 6/9 = 2/3

∠B = ∠E = 75°

Since the ratio of two sides is the same (2/3) and the included angle is congruent, the triangles are similar by SAS similarity.

Problem 4: Solving for an Unknown Side Using SSS Similarity

Problem: Triangles GHI and JKL are similar. GH = 5 cm, HI = 7 cm, GI = 9 cm, and JK = 10 cm. Find the lengths of KL and JL.

Solution: Since the triangles are similar, their sides are proportional. Let's set up proportions:

GH/JK = HI/KL = GI/JL

5/10 = 7/KL = 9/JL

From 5/10 = 7/KL: 5KL = 70 => KL = 14 cm

From 5/10 = 9/JL: 5JL = 90 => JL = 18 cm

Problem 5: Real-World Application: Shadow Lengths

Problem: A tree casts a shadow of 15 meters. At the same time, a 2-meter tall person casts a shadow of 3 meters. How tall is the tree?

Solution: The tree and the person form similar triangles with the sun's rays. Let's denote the height of the tree as 'h'. We can set up a proportion:

h/2 = 15/3

3h = 30

h = 10 meters. The tree is 10 meters tall.

Problem 6: Complex Proportionality and Unknown Angles

Problem: Triangle ABC is similar to triangle DEF. AB = 12 cm, BC = 18 cm, AC = 24 cm, and DE = 8 cm. Find the lengths of EF and DF, and the measure of ∠F if ∠C = 40°.

Solution: We can set up proportions:

AB/DE = BC/EF = AC/DF

12/8 = 18/EF = 24/DF

Simplifying the first ratio: 3/2

Therefore:

3/2 = 18/EF => EF = 12 cm

3/2 = 24/DF => DF = 16 cm

Since ∠C = ∠F = 40°, ∠F = 40°.

Problem 7: Indirect Measurement Using Similar Triangles

Problem: You want to measure the width of a river. You stand at point A, directly across from a tree at point B on the other side of the river. You walk 20 meters along the riverbank to point C. Then, you turn and walk until your line of sight to the tree at B makes a right angle with AC at point C. You measure the distance from C to your new position, D, and find it to be 15 meters. How wide is the river?

Solution: Triangles ABC and BCD are similar right-angled triangles. We can set up a proportion:

AB/BC = BC/CD

AB/20 = 20/15

15AB = 400

AB = 400/15 = 80/3 ≈ 26.67 meters. The river is approximately 26.67 meters wide.

Conclusion: Mastering Similar Triangles

Understanding and applying similar triangle theorems allows us to solve a wide range of geometric problems, from basic proportionality calculations to complex real-world applications. The key is to carefully identify corresponding angles and sides, set up accurate proportions, and apply the appropriate similarity criterion (AAA, SAS, or SSS). Through consistent practice and a solid understanding of these principles, you'll confidently tackle any similar triangle problem you encounter. Remember to always check your work and ensure your solutions are logically sound and mathematically accurate. Consistent practice is the key to mastering this crucial geometric concept.