Question

1. what is the value of y in the diagram below?

• (2sqrt{3})
• (7sqrt{2})
• (sqrt{14})
• (sqrt{6})

Explanation:

🆕 New Concept Discovered: Geometric Mean Theorems
Using relationships in right triangles split by an altitude.

Step 1: Identify the geometric relationship

The diagram shows a large right triangle split into two smaller right triangles by an altitude drawn to the hypotenuse.

According to the Geometric Mean Theorem (Leg Rule), the length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

• Leg: \( y \)
• Adjacent hypotenuse segment: \( 2 \)
• Entire hypotenuse: \( 2 + 5 = 7 \)

Step 2: Set up the equation

Using the geometric mean relationship:

$$ \frac{\text{entire hypotenuse}}{\text{leg}} = \frac{\text{leg}}{\text{adjacent segment}} $$

Substitute the known values:

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

Step 3: Solve for y

Multiply both sides to solve for \( y^2 \):

$$ y^2 = 7 \cdot 2 $$

$$ y^2 = 14 $$

$$ y = \sqrt{14} $$

Answer:

\( \sqrt{14} \)