Alright, let's analyze the problem step by step based on the given data and its implications.
### Given Information:
- The length of the ladder (hypotenuse) is 12 feet.
- The distance from the base of the wall to the base of the ladder (one leg of the triangle) is [tex]\(6\sqrt{2}\)[/tex] feet.
### Calculations and Deductions:
1. Using the Pythagorean Theorem:
- Since this is a right triangle, we use the Pythagorean theorem, which states that:
[tex]\[
\text{(hypotenuse)}^2 = \text{(base)}^2 + \text{(height)}^2
\][/tex]
- Given values:
[tex]\[
\text{hypotenuse} = 12 \text{ feet}
\][/tex]
[tex]\[
\text{base} = 6\sqrt{2} \text{ feet}
\][/tex]
- Calculating the hypotenuse squared:
[tex]\[
12^2 = 144
\][/tex]
- Calculating the base squared:
[tex]\[
(6\sqrt{2})^2 = 6^2 \cdot 2 = 36 \cdot 2 = 72
\][/tex]
- Using the Pythagorean theorem to find the height:
[tex]\[
\text{height}^2 = 144 - 72 = 72
\][/tex]
[tex]\[
\text{height} = \sqrt{72} = 6\sqrt{2} \text{ feet}
\][/tex]
2. Isosceles Triangle Check:
- To be an isosceles triangle, at least two sides should be of equal length.
- Here, the base is [tex]\(6\sqrt{2} \)[/tex] feet and the height is also [tex]\(6\sqrt{2} \)[/tex] feet, but the hypotenuse is 12 feet.
- Since the hypotenuse is not equal to the base or height, the triangle is not isosceles.
3. Leg-to-Hypotenuse Ratio:
- For the base-leg to hypotenuse ratio:
[tex]\[
\text{Ratio} = \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2}
\][/tex]
- This simplifies to:
[tex]\[
1 : \frac{\sqrt{2}}{2}
\][/tex]
- For the height-leg to hypotenuse ratio (since the height equals the base):
[tex]\[
\text{Ratio} = \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2}
\][/tex]
- This also simplifies to:
[tex]\[
1 : \frac{\sqrt{2}}{2}
\][/tex]
4. Non-right Angles Congruency:
- In a right triangle, if the two legs (base and height) are of equal length, the non-right angles must both be 45 degrees.
- Since the base and height are both [tex]\(6\sqrt{2}\)[/tex] feet, the non-right angles are indeed congruent.
5. Ladder as the Longest Length:
- The hypotenuse (ladder) is 12 feet, and the other sides (base and height) are each [tex]\(6\sqrt{2}\)[/tex] feet, approximately 8.49 feet.
- Thus, the ladder is indeed the longest side in the triangle.
### Conclusion:
Given the information:
- The triangle is not isosceles.
- The leg-to-hypotenuse ratio is [tex]\(\frac{\sqrt{2}}{2} : 1 = 1: \frac{\sqrt{2}}{2}\)[/tex].
- The non-right angles are congruent.
- The ladder represents the longest length in the triangle.
Thus, the conclusions we can verify are:
1. The leg-to-hypotenuse ratio is [tex]\(1: \frac{\sqrt{2}}{2}\)[/tex].
2. The non-right angles are congruent.
3. The ladder represents the longest length in the triangle.
