Answer :
Let's start by understanding the given information and the geometric setup of the problem:
1. Lengths Given:
- The ladder length (hypotenuse) is 12 feet.
- The distance from the base of the wall to the base of the ladder is [tex]\(6 \sqrt{2}\)[/tex] feet.
We need to analyze the properties of the triangle formed by the ground (base), wall (height), and ladder (hypotenuse).
2. Height of the Wall:
Using the Pythagorean theorem:
[tex]\[ \text{hypotenuse}^2 = \text{base}^2 + \text{height}^2 \][/tex]
Given:
- Hypotenuse = 12 feet
- Base = [tex]\(6\sqrt{2}\)[/tex] feet
Substitute these values into the equation:
[tex]\[ 12^2 = (6\sqrt{2})^2 + \text{height}^2 \][/tex]
[tex]\[ 144 = 72 + \text{height}^2 \][/tex]
Solving for height:
[tex]\[ \text{height}^2 = 144 - 72 \][/tex]
[tex]\[ \text{height}^2 = 72 \][/tex]
[tex]\[ \text{height} = \sqrt{72} = 6\sqrt{2} \approx 8.49 \text{ feet} \][/tex]
3. Checking Triangle Properties:
- Isosceles Triangle:
The triangle is isosceles if two sides are of equal length. In this case:
[tex]\[ \text{Base} \neq \text{Height} \quad (6\sqrt{2} \neq 6\sqrt{2}) \][/tex]
Therefore, the triangle is not isosceles.
- Leg-to-Hypotenuse Ratios:
- Ratio 1: The leg-to-hypotenuse ratio [tex]\(1:\sqrt{2}\)[/tex] means:
[tex]\[ \frac{\text{leg}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} \][/tex]
Checking with given base:
[tex]\[ \frac{6\sqrt{2}}{12} = \frac{1}{2} \neq \frac{1}{\sqrt{2}} \][/tex]
- The leg-to-hypotenuse ratio is not [tex]\(1:\sqrt{2}\)[/tex].
- Ratio 2: The leg-to-hypotenuse ratio [tex]\(1:\frac{\sqrt{2}}{2}\)[/tex] means:
[tex]\[ \frac{\text{leg}}{\text{hypotenuse}} = \frac{1}{\left(\frac{\sqrt{2}}{2}\right)} = \frac{\frac{\sqrt{2}}{2}} = \sqrt{2} \][/tex]
Checking with given base:
[tex]\[ \frac{6\sqrt{2}}{12} = \frac{1}{2} \neq \sqrt{2} \][/tex]
- The leg-to-hypotenuse ratio is not [tex]\(1:\frac{\sqrt{2}}{2}\)[/tex].
- Nonright Angles Congruent:
Nonright angles are congruent if the triangle is isosceles. Since our triangle is not isosceles, the nonright angles are not congruent.
- Ladder as the Longest Length:
The ladder, being the hypotenuse, is always the longest side in a right-angled triangle.
Hence, the ladder represents the longest length in the triangle.
4. Conclusion:
Based on the analysis above, the correct conclusions about the triangle are:
- The triangle is not isosceles.
- The leg-to-hypotenuse ratio is neither [tex]\(1:\sqrt{2}\)[/tex] nor [tex]\(1:\frac{\sqrt{2}}{2}\)[/tex].
- The nonright angles are not congruent.
- The ladder represents the longest length in the triangle.
So, the applicable property is:
- The ladder represents the longest length in the triangle.
1. Lengths Given:
- The ladder length (hypotenuse) is 12 feet.
- The distance from the base of the wall to the base of the ladder is [tex]\(6 \sqrt{2}\)[/tex] feet.
We need to analyze the properties of the triangle formed by the ground (base), wall (height), and ladder (hypotenuse).
2. Height of the Wall:
Using the Pythagorean theorem:
[tex]\[ \text{hypotenuse}^2 = \text{base}^2 + \text{height}^2 \][/tex]
Given:
- Hypotenuse = 12 feet
- Base = [tex]\(6\sqrt{2}\)[/tex] feet
Substitute these values into the equation:
[tex]\[ 12^2 = (6\sqrt{2})^2 + \text{height}^2 \][/tex]
[tex]\[ 144 = 72 + \text{height}^2 \][/tex]
Solving for height:
[tex]\[ \text{height}^2 = 144 - 72 \][/tex]
[tex]\[ \text{height}^2 = 72 \][/tex]
[tex]\[ \text{height} = \sqrt{72} = 6\sqrt{2} \approx 8.49 \text{ feet} \][/tex]
3. Checking Triangle Properties:
- Isosceles Triangle:
The triangle is isosceles if two sides are of equal length. In this case:
[tex]\[ \text{Base} \neq \text{Height} \quad (6\sqrt{2} \neq 6\sqrt{2}) \][/tex]
Therefore, the triangle is not isosceles.
- Leg-to-Hypotenuse Ratios:
- Ratio 1: The leg-to-hypotenuse ratio [tex]\(1:\sqrt{2}\)[/tex] means:
[tex]\[ \frac{\text{leg}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} \][/tex]
Checking with given base:
[tex]\[ \frac{6\sqrt{2}}{12} = \frac{1}{2} \neq \frac{1}{\sqrt{2}} \][/tex]
- The leg-to-hypotenuse ratio is not [tex]\(1:\sqrt{2}\)[/tex].
- Ratio 2: The leg-to-hypotenuse ratio [tex]\(1:\frac{\sqrt{2}}{2}\)[/tex] means:
[tex]\[ \frac{\text{leg}}{\text{hypotenuse}} = \frac{1}{\left(\frac{\sqrt{2}}{2}\right)} = \frac{\frac{\sqrt{2}}{2}} = \sqrt{2} \][/tex]
Checking with given base:
[tex]\[ \frac{6\sqrt{2}}{12} = \frac{1}{2} \neq \sqrt{2} \][/tex]
- The leg-to-hypotenuse ratio is not [tex]\(1:\frac{\sqrt{2}}{2}\)[/tex].
- Nonright Angles Congruent:
Nonright angles are congruent if the triangle is isosceles. Since our triangle is not isosceles, the nonright angles are not congruent.
- Ladder as the Longest Length:
The ladder, being the hypotenuse, is always the longest side in a right-angled triangle.
Hence, the ladder represents the longest length in the triangle.
4. Conclusion:
Based on the analysis above, the correct conclusions about the triangle are:
- The triangle is not isosceles.
- The leg-to-hypotenuse ratio is neither [tex]\(1:\sqrt{2}\)[/tex] nor [tex]\(1:\frac{\sqrt{2}}{2}\)[/tex].
- The nonright angles are not congruent.
- The ladder represents the longest length in the triangle.
So, the applicable property is:
- The ladder represents the longest length in the triangle.
