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Sagot :
Let's analyze the given scenario step-by-step to understand the properties of the triangle formed by the ground, wall, and ladder.
### Step 1: Determine the Height of the Wall
You have a right-angled triangle where:
- The ladder forms the hypotenuse, which is 12 feet.
- The distance from the base of the wall to the base of the ladder is [tex]\(6 \sqrt{2}\)[/tex] feet.
Using the Pythagorean theorem:
[tex]\[ \text{hypotenuse}^2 = \text{base}^2 + \text{height}^2 \][/tex]
[tex]\[ 12^2 = (6\sqrt{2})^2 + \text{height}^2 \][/tex]
[tex]\[ 144 = 72 + \text{height}^2 \][/tex]
[tex]\[ \text{height}^2 = 72 \][/tex]
[tex]\[ \text{height} = \sqrt{72} = 6\sqrt{2} \approx 8.485 \, \text{feet} \][/tex]
### Step 2: Verify if the Triangle is Isosceles
An isosceles triangle has two sides of equal length. Comparing the base and height:
- Base: [tex]\(6\sqrt{2}\)[/tex] feet
- Height: [tex]\(6\sqrt{2}\)[/tex] feet
Since the base and height are both [tex]\(6\sqrt{2}\)[/tex] feet, the triangle is not isosceles because the hypotenuse (ladder) is distinctly longer at 12 feet.
### Step 3: Determine the Leg-to-Hypotenuse Ratios
To find out the leg-to-hypotenuse ratio, compare each leg (base and height) to the hypotenuse:
- The base ([tex]\(6\sqrt{2}\)[/tex] feet) to hypotenuse (12 feet):
[tex]\[ \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2} \approx 0.707 \][/tex]
- The height ([tex]\(6\sqrt{2}\)[/tex] feet) to hypotenuse (12 feet):
[tex]\[ \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2} \approx 0.707 \][/tex]
So, the leg-to-hypotenuse ratio is approximately [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
### Step 4: Check if the Nonright Angles are Congruent
For non-right angles to be congruent, the triangle must be isosceles with equal sides forming these angles. Since the triangle is right-angled but not isosceles, the non-right angles are not equal.
### Step 5: Identify the Longest Length
In a right-angled triangle, the hypotenuse is always the longest side. Here, the hypotenuse is the ladder, which is 12 feet.
### Conclusion
From the analysis:
1. The triangle is not isosceles.
2. The leg-to-hypotenuse ratio is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
3. The nonright angles are not congruent.
4. The ladder represents the longest length in the triangle.
Therefore, based on the information provided and the steps above, the correct statements about the triangle are:
- The leg-to-hypotenuse ratio is [tex]\(1: \frac{\sqrt{2}}{2}\)[/tex].
- The ladder represents the longest length in the triangle.
### Step 1: Determine the Height of the Wall
You have a right-angled triangle where:
- The ladder forms the hypotenuse, which is 12 feet.
- The distance from the base of the wall to the base of the ladder is [tex]\(6 \sqrt{2}\)[/tex] feet.
Using the Pythagorean theorem:
[tex]\[ \text{hypotenuse}^2 = \text{base}^2 + \text{height}^2 \][/tex]
[tex]\[ 12^2 = (6\sqrt{2})^2 + \text{height}^2 \][/tex]
[tex]\[ 144 = 72 + \text{height}^2 \][/tex]
[tex]\[ \text{height}^2 = 72 \][/tex]
[tex]\[ \text{height} = \sqrt{72} = 6\sqrt{2} \approx 8.485 \, \text{feet} \][/tex]
### Step 2: Verify if the Triangle is Isosceles
An isosceles triangle has two sides of equal length. Comparing the base and height:
- Base: [tex]\(6\sqrt{2}\)[/tex] feet
- Height: [tex]\(6\sqrt{2}\)[/tex] feet
Since the base and height are both [tex]\(6\sqrt{2}\)[/tex] feet, the triangle is not isosceles because the hypotenuse (ladder) is distinctly longer at 12 feet.
### Step 3: Determine the Leg-to-Hypotenuse Ratios
To find out the leg-to-hypotenuse ratio, compare each leg (base and height) to the hypotenuse:
- The base ([tex]\(6\sqrt{2}\)[/tex] feet) to hypotenuse (12 feet):
[tex]\[ \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2} \approx 0.707 \][/tex]
- The height ([tex]\(6\sqrt{2}\)[/tex] feet) to hypotenuse (12 feet):
[tex]\[ \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2} \approx 0.707 \][/tex]
So, the leg-to-hypotenuse ratio is approximately [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
### Step 4: Check if the Nonright Angles are Congruent
For non-right angles to be congruent, the triangle must be isosceles with equal sides forming these angles. Since the triangle is right-angled but not isosceles, the non-right angles are not equal.
### Step 5: Identify the Longest Length
In a right-angled triangle, the hypotenuse is always the longest side. Here, the hypotenuse is the ladder, which is 12 feet.
### Conclusion
From the analysis:
1. The triangle is not isosceles.
2. The leg-to-hypotenuse ratio is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
3. The nonright angles are not congruent.
4. The ladder represents the longest length in the triangle.
Therefore, based on the information provided and the steps above, the correct statements about the triangle are:
- The leg-to-hypotenuse ratio is [tex]\(1: \frac{\sqrt{2}}{2}\)[/tex].
- The ladder represents the longest length in the triangle.
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