Question

what is the unknown side length?

Explanation:

🆕 New Concept Discovered: Triangle Proportionality Theorem
Using parallel lines to find missing lengths in triangles.

Step 1: Identify similar triangles

The red arrows indicate that the two horizontal line segments are parallel. Because these lines are parallel, the corresponding angles are equal, which makes the small top triangle similar to the entire large triangle by Angle-Angle (AA) similarity.

Step 2: Set up the proportion

For similar triangles, the ratio of corresponding side lengths is equal.

Let \( x \) be the unknown total length of the long side on the left (indicated by the blue arrow with the question mark).

The corresponding side of the small triangle has a length of \( 4 \).

The right side of the small triangle has a length of \( 2 \).
The total right side of the large triangle is:

$$ 2 + 5 = 7 $$

We can set up the ratio of the left side to the right side for both triangles:

$$ \frac{x}{7} = \frac{4}{2} $$

Step 3: Solve for the unknown length

Simplify the fraction on the right side:

$$ \frac{4}{2} = 2 $$

Now, solve for \( x \):

$$ \frac{x}{7} = 2 $$

$$ x = 2 \cdot 7 $$

$$ x = 14 $$

Answer:

14