https://file.aiquickdraw.com/mathbot/file/17295309211048arlxikl.jpg

The image shows a right triangle that is part of a bridge structure. You are asked to find:

1. The distance across the bridge, denoted as \( x \).
2. The distance between the perpendicular beams, denoted as \( y \).

Given information:
- The two equal sides of the triangle are each \( 4.6 \, \text{m} \).
- The triangle has a right angle of \( 90^\circ \), meaning it is an isosceles right triangle.

### Step 1: Calculate \( x \) (the hypotenuse)
Since this is an isosceles right triangle (the two legs are equal), we can apply the Pythagorean Theorem to find \( x \), where:

\[
x^2 = (4.6)^2 + (4.6)^2
\]

\[
x^2 = 21.16 + 21.16 = 42.32
\]

\[
x = \sqrt{42.32} \approx 6.5 \, \text{m}
\]

### Step 2: Calculate \( y \) (the distance between perpendicular beams)
For isosceles right triangles, the legs (which form the right angle) are of equal length, and the relationship between the hypotenuse and the legs follows the formula:

\[
y = \frac{x}{\sqrt{2}}
\]

Substituting \( x = 6.5 \):

\[
y = \frac{6.5}{\sqrt{2}} \approx 4.6 \, \text{m}
\]

### Final Results:
- The distance across the bridge, \( x \), is approximately \( 6.5 \, \text{m} \).
- The distance between the perpendicular beams, \( y \), is approximately \( 4.6 \, \text{m} \).

Would you like further details on any part of this calculation or additional questions?

#### Follow-up Questions:
1. What is the relationship between the legs of an isosceles right triangle and its hypotenuse?
2. How does the Pythagorean Theorem apply to different types of triangles?
3. Can you use trigonometric ratios to solve for \( x \) and \( y \) in a non-right triangle?
4. What other real-life structures use isosceles right triangles for stability?
5. How would the dimensions change if the angle was different from \( 90^\circ \)?

#### Tip:
For right triangles with equal legs, the hypotenuse is always the length of one leg times \( \sqrt{2} \).

The bridge shown is constructed with a pattern of triangles. Using the information in the picture, find the distance across the bridge, x, and the distance between the perpendicular beams, y. Round to two decimal places, if necessary.

Learn how to find the distance across a bridge using geometry and the Pythagorean Theorem. In this example, two equal sides of an isosceles right triangle measure 4.6 meters. We'll calculate the hypotenuse (bridge length) and the distance between perpendicular beams, rounded to two decimal places.

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