Back To Course

Math 102: College Mathematics15 chapters | 122 lessons | 13 flashcard sets

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*DaQuita Hester*

DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.

Similar triangles are used to solve problems in everyday situations. Learn how to solve with similar triangles here, and then test your understanding with a quiz.

Sarah is standing outside next to a flagpole. The sun casts a 4 ft. shadow of Sarah and a 7 ft. shadow of the flagpole. If Sarah is 5 ft. tall, how tall is the flagpole?

Without a ladder and measuring stick, you may think that solving this problem is impossible. However, finding the solution may be easier than you think. For this problem, and others like it, solving becomes a matter of **similar triangles**.

In another lesson, we learned that similar triangles have congruent corresponding angles and proportional corresponding sides and can be quickly identified by the **Angle-Angle (AA)**, **Side-Angle-Side (SAS)**, and **Side-Side-Side (SSS)** similarity theorems. To solve with similar triangles, we will use their side lengths to set up proportions, meaning that we will create fractions for the corresponding sides and set them equal to each other. Let's begin our practice with some basic examples.

Take a look at triangle ABE and triangle ACD above. From the picture, we see that *Angle B* is congruent to *Angle C* and *Angle E* is congruent to *Angle D*. Therefore, these triangles are similar by AA. Let's set up our proportions.

By going in the same order for each fraction, our beginning proportion is 8/5 = *x*/6. To get *x* by itself, we must cross-multiply, which will give us 5*x* = 48. Next, we will divide both sides by 5 to conclude that *x* = 9.6.

Here's another one: This time, we have triangle MOQ and triangle NOP. Once again, based on the congruent marks in the picture above, we know that these two triangles are similar by AA. So, let's start the problem.

In the figure, we were given the length of side MO, meaning that we also need the length of side QO for the proportion. But, we were not given this information, so we must calculate it.

To get the length of QO, we must add QP + PO since these two pieces form segment QO. By doing this, we add 7 + 8 to see that QO = 15. Now, we are ready for our proportion.

By going in the same order, our proportion is 7/15 = *x*/18. Then, when we cross-multiply, we see that 15*x* = 126. Once we divide both sides by 15, we can conclude that *x* = 8.4.

Now that we've covered some of the basics, let's do some real-world examples, starting with Sarah and the flagpole. Recall that Sarah is 5 ft. tall and has a 4 ft. shadow, and we are looking for the height of the flagpole, which has a 7 ft. shadow.

Do you see the similar triangles above? Sarah and the flagpole are the vertical legs and the shadows are the horizontal legs. Connecting the end of the shadows to the top of Sarah and the flagpole complete the triangles. Since the sun is casting both shadows, and since Sarah is standing right beside the flagpole, the top angles in both triangles are congruent. Additionally, since Sarah and the flagpole are perpendicular with the ground, each triangle has a right angle. Therefore, these triangles are similar by the AA similarity theorem.

Now we can write a proportion. Using *x* to represent the height of the flagpole, we begin with 5/4 = *x*/7. After cross-multiplying, we have 4*x* = 35, and then, by dividing both sides by 4, we see that *x* = 8.75. Therefore, the flagpole has a height of 8.75 ft.

Let's try another: Julie places a mirror on the ground 30 ft. away from the base of her office building and walks backwards until she can see the top of the building in the mirror. If Julie is 5.5 ft. tall and is standing 5ft. away from the mirror, how tall is the building?

For this scenario, Julie and the building serve as the vertical legs of the triangles and the distances from the mirror to Julie and the building are the horizontal legs. The triangles are completed by connecting the mirror to the top of both Julie and the building.

Since Julie can see the top of her building through the mirror, the two angles formed at the mirror are congruent. Also, Julie and her office building form right angles with the ground, showing us that these two triangles above are also similar by AA. Now, it's time for the proportion.

By letting *x* represent the height of the building, our beginning proportion is *x*/30 = 5.5/5. From cross-multiplying, we get 5*x* = 165. Then, by dividing both sides by 5, we can conclude that *x* = 33. Therefore, the height of Julie's office building is 33 ft.

In conclusion, **similar triangles** can be applied to solve everyday, real-world problems. Since the triangles are similar, solving with them requires us to set up proportions that compare their corresponding sides. After going in the same order to set up the proportions, cross-multiplying is necessary to complete and solve the problem.

You'll be able to apply the angle-angle theorem of similar triangles to solve real-world problems after watching this video lesson.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
8 in chapter 14 of the course:

Back To Course

Math 102: College Mathematics15 chapters | 122 lessons | 13 flashcard sets

- Go to Logic

- Go to Sets

- Properties of Shapes: Rectangles, Squares and Rhombuses 5:46
- Properties of Shapes: Triangles 5:09
- Perimeter of Triangles and Rectangles 8:54
- Area of Triangles and Rectangles 5:43
- Circles: Area and Circumference 8:21
- The Pythagorean Theorem: Practice and Application 7:33
- How to Identify Similar Triangles 7:23
- Applications of Similar Triangles 6:23
- Types of Angles: Vertical, Corresponding, Alternate Interior & Others 10:28
- Angles and Triangles: Practice Problems 7:43
- Properties of Shapes: Circles 4:45
- Go to Geometry

- Computer Science 109: Introduction to Programming
- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Algorithmic Analysis, Sorting & Searching
- Computer Programming Basics
- Stacks & Queues for Data Structures
- Functions & Modules in Programming
- Built-In Data Types for Programming
- CEOE Test Cost
- PHR Exam Registration Information
- Claiming a Tax Deduction for Your Study.com Teacher Edition
- What is the PHR Exam?
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP

- Risk Identification: Process & Analysis
- Design Constraints in Engineering: Definition & Example
- Synagogue vs. Temple: Definitions & Differences
- A Midsummer Night's Dream Play Within a Play: Analysis
- What is a VLAN? - Definition, Function & Implementation
- Room 101 Lesson Plan
- Story Quilt Lesson Plan
- Quiz & Worksheet - Narrative of the Life of Frederick Douglass
- Quiz & Worksheet - What is Process Synchronization in OS?
- Quiz & Worksheet - Dorsiflexion vs. Plantar Flexion
- Quiz & Worksheet - Kinds of Plastic & Their Uses
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Health and Physical Education Lesson Plans
- Cyberbullying Facts & Resources for Teachers

- AP Biology: Help and Review
- Chemistry: High School
- Physical Geology for Teachers: Professional Development
- U.S. History II: Certificate Program
- Introduction to Music: Certificate Program
- Managing the Employer-Worker Relationship
- Density and Buoyancy
- Quiz & Worksheet - Attributes of Gothic Figural Art
- Quiz & Worksheet - The USSR in WWII
- Quiz & Worksheet - Abstract Expressionist Sculptors
- Quiz & Worksheet - Contemporary Acrylic vs. Oil Painting
- Quiz & Worksheet - MRP Process Steps

- Participating in Clinical Studies: Considerations & Eligibility
- Student & Professional Organizations in Marketing Education
- Best Psychology Books for Undergraduates
- How to Pass Actuarial Exams
- Free Online Accounting Courses with a Certificate
- Global History & Geography Regents Exam Info
- Is PHP Hard to Learn?
- States of Matter Experiments for Kids
- Grants for Field Trips
- What is the NYS Regents Scholarship?
- 7th Grade Summer Reading List
- How to Ace Your Job Interview

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject