ACT Math: Plane Geometry - Similar Triangles, AA, SAS, SSS, Ratios of Sides

What This Is and Why It Matters for the ACT

Plane Geometry, specifically Similar Triangles, is a crucial topic that appears in the Math section of the ACT. You'll see it on almost every Math test, and it's considered an intermediate-level topic. Be prepared to apply concepts like AA, SAS, SSS, and ratios of sides to solve problems.

Key Concepts (What You Must Know)

• Definition: Similar triangles are triangles that have the same shape but not necessarily the same size.
• AA (Angle-Angle) Similarity: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
• SAS (Side-Angle-Side) Similarity: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are similar.
• SSS (Side-Side-Side) Similarity: If three sides of one triangle are congruent to three sides of another triangle, the triangles are similar.
• Ratios of Sides: The ratio of the lengths of corresponding sides of similar triangles is the same.

Step-by-Step Strategy for This Topic

1. Read the problem carefully: Identify the type of similarity (AA, SAS, SSS) and the given information.
2. Draw a diagram: Sketch the triangles to visualize the problem and identify any congruent parts.
3. Apply the similarity concept: Use the given information to determine the type of similarity and the ratio of the sides.
4. Eliminate wrong answers: Check each option to see if it's consistent with the similarity concept.
5. Check your work: Verify that your answer is consistent with the given information and the similarity concept.
6. Time management: Allocate 1-2 minutes per problem, depending on your speed and accuracy.

How It’s Tested on the ACT

• Math: Multiple-choice questions with five answer choices, often involving diagram-based problems.
• Common distractors: Overlooking the given information, Misinterpreting the similarity concept, and Not checking the answer choices.

Common Mistakes & Exam Traps

• The mistake: Overlooking the given information.
• Why it happens: Misreading the problem or rushing through the question.
• How to avoid it: Carefully read the problem and identify the given information before drawing a diagram.
• Exam board insight: The examiners will penalize you for not following the instructions carefully.
• The mistake: Misinterpreting the similarity concept.
• Why it happens: Confusing the different types of similarity or not understanding the concept.
• How to avoid it: Review the similarity concepts carefully and practice applying them to different problems.
• Exam board insight: The examiners will expect you to apply the similarity concept correctly.
• The mistake: Not checking the answer choices.
• Why it happens: Rushing through the question or not verifying the answer.
• How to avoid it: Check each answer choice carefully and verify that it's consistent with the similarity concept.
• Exam board insight: The examiners will penalize you for not checking the answer choices carefully.

Practice Questions (3-5 questions)

Question 1

In the figure below, triangle ABC is similar to triangle DEF. If the ratio of the lengths of corresponding sides is 2:3, what is the length of side EF?

[Insert diagram]

A) 4 B) 6 C) 12 D) 18 E) 24

Answer: C) 12

Explanation: The ratio of the lengths of corresponding sides is 2:3, so the length of side EF is 3 times the length of side AB. Since the length of side AB is not given, we can let it be x. Then, the length of side EF is 3x. We can find the length of side AB by using the ratio of the lengths of corresponding sides, which is 2:3. Since the ratio of the lengths of corresponding sides is 2:3, the length of side AB is 2x. Now, we can substitute 2x for the length of side AB in the equation for the length of side EF: 3(2x) = 6x. Since the length of side EF is 6x, the length of side EF is 6 times the length of side AB, which is 6(2x) = 12x.

Question 2

In the figure below, triangle ABC is similar to triangle DEF. If the length of side AB is 6 and the length of side EF is 12, what is the ratio of the lengths of corresponding sides?

[Insert diagram]

A) 1:2 B) 2:3 C) 1:2 D) 3:4 E) 4:5

Answer: C) 1:2

Explanation: Since the length of side AB is 6 and the length of side EF is 12, the ratio of the lengths of corresponding sides is 6:12, which simplifies to 1:2.

Question 3

In the figure below, triangle ABC is similar to triangle DEF. If the length of side AB is 8 and the length of side EF is 16, what is the length of side BC?

[Insert diagram]

A) 4 B) 6 C) 8 D) 10 E) 12

Answer: C) 8

Explanation: Since the length of side AB is 8 and the length of side EF is 16, the ratio of the lengths of corresponding sides is 8:16, which simplifies to 1:2. Since the ratio of the lengths of corresponding sides is 1:2, the length of side BC is half the length of side AB, which is 8/2 = 4. However, this is not an option. Since the ratio of the lengths of corresponding sides is 1:2, the length of side BC is also half the length of side EF, which is 16/2 = 8.

Quick Reference Card (60-Second Summary)

• AA Similarity: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
• SAS Similarity: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are similar.
• SSS Similarity: If three sides of one triangle are congruent to three sides of another triangle, the triangles are similar.
• Ratios of Sides: The ratio of the lengths of corresponding sides of similar triangles is the same.
• Similar Triangles: Similar triangles have the same shape but not necessarily the same size.

If You Get Stuck on Test Day

• What to do when you don't know the answer: Eliminate any obviously incorrect options and make an educated guess.
• Pacing strategy: Allocate 1-2 minutes per problem, depending on your speed and accuracy.
• When to skip and come back: If you're stuck on a problem, skip it and come back to it later. You can also use the process of elimination to eliminate any obviously incorrect options.

Related ACT Topics

• Congruent Triangles: Congruent triangles have the same size and shape.
• Perimeter and Area: The perimeter and area of similar triangles are proportional to the square of the ratio of their corresponding sides.
• Right Triangles: Right triangles have one right angle (90 degrees).