Answer :
Given the ladder length (hypotenuse) is 12 feet and the base length (one leg of the triangle) is [tex]\(6\sqrt{2}\)[/tex] feet, let's analyze the properties of the triangle formed by these measurements.
### Step-by-Step Solution:
1. Calculate the Height of the Wall:
- Using the Pythagorean theorem [tex]\( a^2 + b^2 = c^2 \)[/tex], where:
- [tex]\( c \)[/tex] = ladder length = 12 feet,
- [tex]\( b \)[/tex] = base length = [tex]\(6\sqrt{2}\)[/tex] feet,
- [tex]\( a \)[/tex] = the height of the wall.
- Substituting the known values:
[tex]\[ a^2 + (6\sqrt{2})^2 = 12^2 \][/tex]
[tex]\[ a^2 + 72 = 144 \][/tex]
[tex]\[ a^2 = 144 - 72 \][/tex]
[tex]\[ a^2 = 72 \][/tex]
[tex]\[ a = \sqrt{72} = 6\sqrt{2} \][/tex]
- The height of the wall (other leg of the triangle) is [tex]\( 6\sqrt{2} \)[/tex] feet.
2. Determine if the Triangle is Isosceles:
- A triangle is isosceles if at least two sides are equal.
- Here, the base length is [tex]\(6\sqrt{2}\)[/tex] and the height is [tex]\(6\sqrt{2}\)[/tex], which are equal.
- Therefore, the triangle formed is not isosceles, since both legs are equal but the hypotenuse is different from both.
3. Check the Leg-to-Hypotenuse Ratio:
- The ratio of one leg to the hypotenuse:
[tex]\[ \text{Ratio} = \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2} \][/tex]
- This simplifies to [tex]\( \frac{\sqrt{2}}{2} \)[/tex] or [tex]\( 1 : \frac{\sqrt{2}}{2} \)[/tex].
4. Check Congruence of Nonright Angles:
- In an isosceles right triangle (45°-45°-90°), the nonright angles are congruent.
- Since the triangle here has two equal legs ([tex]\(6\sqrt{2}\)[/tex] feet each), it should be a 45°-45°-90° triangle. However, this does not hold since the hypotenuse cannot be matched to [tex]\(6\sqrt{2}\)[/tex] if it truly were this type of triangle.
5. Longest Length in the Triangle:
- The ladder represents the hypotenuse of the right triangle.
- In any right triangle, the hypotenuse is the longest side compared to the legs.
### Conclusion:
Given the information:
- The triangle is not isosceles.
- The triangle does not have a leg-to-hypotenuse ratio of [tex]\(1 : \sqrt{2}\)[/tex].
- The triangle fits the ratio [tex]\(1 : \frac{\sqrt{2}}{2}\)[/tex].
- The nonright angles are not congruent as the triangle is confirmed to be not isosceles.
- The ladder is the longest length in the triangle because it represents the hypotenuse.
Therefore, the accurate statements based on the above solution are:
- The leg-to-hypotenuse ratio is [tex]\(1: \frac{\sqrt{2}}{2}\)[/tex]
- The ladder represents the longest length in the triangle
### Step-by-Step Solution:
1. Calculate the Height of the Wall:
- Using the Pythagorean theorem [tex]\( a^2 + b^2 = c^2 \)[/tex], where:
- [tex]\( c \)[/tex] = ladder length = 12 feet,
- [tex]\( b \)[/tex] = base length = [tex]\(6\sqrt{2}\)[/tex] feet,
- [tex]\( a \)[/tex] = the height of the wall.
- Substituting the known values:
[tex]\[ a^2 + (6\sqrt{2})^2 = 12^2 \][/tex]
[tex]\[ a^2 + 72 = 144 \][/tex]
[tex]\[ a^2 = 144 - 72 \][/tex]
[tex]\[ a^2 = 72 \][/tex]
[tex]\[ a = \sqrt{72} = 6\sqrt{2} \][/tex]
- The height of the wall (other leg of the triangle) is [tex]\( 6\sqrt{2} \)[/tex] feet.
2. Determine if the Triangle is Isosceles:
- A triangle is isosceles if at least two sides are equal.
- Here, the base length is [tex]\(6\sqrt{2}\)[/tex] and the height is [tex]\(6\sqrt{2}\)[/tex], which are equal.
- Therefore, the triangle formed is not isosceles, since both legs are equal but the hypotenuse is different from both.
3. Check the Leg-to-Hypotenuse Ratio:
- The ratio of one leg to the hypotenuse:
[tex]\[ \text{Ratio} = \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2} \][/tex]
- This simplifies to [tex]\( \frac{\sqrt{2}}{2} \)[/tex] or [tex]\( 1 : \frac{\sqrt{2}}{2} \)[/tex].
4. Check Congruence of Nonright Angles:
- In an isosceles right triangle (45°-45°-90°), the nonright angles are congruent.
- Since the triangle here has two equal legs ([tex]\(6\sqrt{2}\)[/tex] feet each), it should be a 45°-45°-90° triangle. However, this does not hold since the hypotenuse cannot be matched to [tex]\(6\sqrt{2}\)[/tex] if it truly were this type of triangle.
5. Longest Length in the Triangle:
- The ladder represents the hypotenuse of the right triangle.
- In any right triangle, the hypotenuse is the longest side compared to the legs.
### Conclusion:
Given the information:
- The triangle is not isosceles.
- The triangle does not have a leg-to-hypotenuse ratio of [tex]\(1 : \sqrt{2}\)[/tex].
- The triangle fits the ratio [tex]\(1 : \frac{\sqrt{2}}{2}\)[/tex].
- The nonright angles are not congruent as the triangle is confirmed to be not isosceles.
- The ladder is the longest length in the triangle because it represents the hypotenuse.
Therefore, the accurate statements based on the above solution are:
- The leg-to-hypotenuse ratio is [tex]\(1: \frac{\sqrt{2}}{2}\)[/tex]
- The ladder represents the longest length in the triangle
