KutaSoftware: Geometry- Similar Right Triangles Part 1 - YouTube - Free Printable

Let’s solve this step by step.

We are given a right triangle with an altitude drawn to the hypotenuse. This creates two smaller right triangles that are similar to each other and to the original triangle — this is called the Geometric Mean Theorem (or Altitude Rule) in similar right triangles.

In the diagram:
- The entire hypotenuse of the big triangle is 100.
- One segment of the hypotenuse (adjacent to the side labeled 36) is 36.
- So, the other segment must be:
→ 100 - 36 = 64

The altitude (labeled x) is drawn from the right angle to the hypotenuse, splitting it into segments of 36 and 64.

According to the Geometric Mean Theorem for altitudes:

> The length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse.

That means:

→ x² = 36 × 64

Let’s calculate that:

First, multiply 36 × 64.

You can break it down:

36 × 64 = 36 × (60 + 4) = (36×60) + (36×4)

36×60 = 2160
36×4 = 144
Add them: 2160 + 144 = 2304

So, x² = 2304

Now take the square root of both sides:

x = √2304

What’s the square root of 2304?

Let’s think:
48 × 48 = ?

40×40 = 1600
8×8 = 64
But better to compute directly:

48 × 48 = (50 - 2)² = 2500 - 200 + 4 = 2304 ✔

So, x = 48

Final Answer:
48