Similar Triangles Inside a Triangle

🔑 Key Concepts

• When a line segment is drawn inside a triangle parallel to one side, it creates a smaller triangle that is similar to the larger triangle.
• Similar triangles have proportional corresponding side lengths.
• The small yellow triangle and the large triangle have the same shape, so their matching sides can be placed in a proportion.
• The full base of the large triangle is \(10+x\), not just \(10\), because the smaller triangle adds an additional length of \(x\).
• Solving the proportion gives the missing value \(x=5\).

✏️ Worked Example

Problem from the video: What is the value of \(x\)?

The smaller yellow triangle is similar to the larger triangle because the vertical segment inside the triangle is parallel to the left side of the large triangle.

Step 1: Set Up the Proportion

The small triangle has a vertical height of \(2\) m.

The large triangle has a vertical height of \(6\) m.

The small triangle has a base of \(x\).

The large triangle has a full base of \(10+x\).

Match corresponding sides:

\(\frac{2}{6}=\frac{x}{10+x}\)

Important: The denominator is \(10+x\) because the full base of the large triangle includes both parts.

Step 2: Solve for \(x\)

\(\frac{2}{6}=\frac{x}{10+x}\)

Cross multiply:

\(6x=2(10+x)\)

Distribute:

\(6x=20+2x\)

Subtract \(2x\) from both sides:

\(4x=20\)

Divide both sides by \(4\):

\(x=5\)

Answer: \(x=5\) meters.

Aha Moment

The small triangle and large triangle are proportional because they are similar. The small height \(2\) compares to the large height \(6\), and the small base \(x\) compares to the large base \(10+x\). This is why the proportion \(\frac{2}{6}=\frac{x}{10+x}\) works.

🧠 Math Vocabulary

• Similar Triangles: Triangles that have the same shape and proportional corresponding side lengths.
• Corresponding Sides: Sides in similar figures that match each other based on position.
• Proportion: An equation showing that two ratios are equal.
• Scale Factor: A multiplier that compares corresponding side lengths in similar figures.
• Parallel Lines: Lines that stay the same distance apart and do not intersect.
• Cross Multiply: A method for solving proportions by multiplying diagonally across equal ratios.
• Variable: A symbol, such as \(x\), that represents an unknown value.

💡 Main Idea

When a segment is drawn inside a triangle parallel to one of its sides, the smaller triangle formed inside the larger triangle is similar to the original triangle. This allows students to set up a proportion using corresponding side lengths and solve for missing values such as \(x\).

📚 What You Should Already Know

Students should know how to identify corresponding sides, write ratios, solve proportions, use cross multiplication, distribute expressions, and solve one-step or two-step equations.

🚀 What Comes Next

After solving similar triangle problems inside larger triangles, students can apply the same proportional reasoning to scale drawings, indirect measurement, maps, blueprints, and more complex similarity word problems.

🧩 Embed This Video in Your LMS

Teachers can embed this YouTube video directly into Canvas, Google Classroom, Schoology, or another learning platform. Click below to show the embed code.