Properties of the Triangle

Oct 13, 2025 by miodeya.com 116 views

Introduction

When a 12-foot ladder is leaning against a wall, it forms a right-angled triangle with the ground and the wall. The distance from the base of the wall to the base of the ladder is given as $6 \sqrt{2}$ feet. This information provides a unique opportunity to explore the properties of the triangle formed by the ground, wall, and ladder. In this article, we will delve into the world of mathematics to determine what can be concluded about this triangle.

The Triangle Formed by the Ground, Wall, and Ladder

The triangle formed by the ground, wall, and ladder is a right-angled triangle, with the ladder being the hypotenuse. The distance from the base of the wall to the base of the ladder is given as $6 \sqrt{2}$ feet, which is the length of one of the legs of the triangle. The length of the ladder, which is the hypotenuse, is given as 12 feet.

Properties of the Triangle

To determine the properties of the triangle, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this can be expressed as:

c² = a² + b²

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Applying the Pythagorean Theorem

In this case, we know the length of the hypotenuse (c = 12 feet) and one of the legs (a = $6 \sqrt{2}$ feet). We can use the Pythagorean theorem to find the length of the other leg (b).

c² = a² + b² 12² = ( $6 \sqrt{2}$ )² + b² 144 = 72 + b² b² = 72 b = √72 b = 6√6

Conclusion

Using the Pythagorean theorem, we have determined that the length of the other leg of the triangle is 6√6 feet. This means that the triangle formed by the ground, wall, and ladder is a 30-60-90 triangle, with the angle opposite the shorter leg being 30 degrees, the angle opposite the longer leg being 60 degrees, and the angle opposite the hypotenuse being 90 degrees.

Properties of a 30-60-90 Triangle

A 30-60-90 triangle is a special type of right-angled triangle that has specific properties. The ratio of the lengths of the sides of a 30-60-90 triangle is 1:√3:2, where the side opposite the 30-degree angle is 1, the side opposite the 60-degree angle is √3, and the hypotenuse is 2.

Verifying the Properties of the Triangle

To verify that the triangle formed by the ground, wall, and ladder is a 30-60-90 triangle, we can check the ratio of the lengths of the sides.

The length of the shorter leg (a) is $6 \sqrt{2}$ feet. The length of the longer leg (b) is 6√6 feet. The length of the hypotenuse (c) is 12 feet.

We can calculate the ratio of the lengths of the sides as follows:

a:b:c = $6 \sqrt{2}$ :6√6:12 = 6√2:6√6:12 = (√2:√6:2) × 6 = 1:√3:2 × 6

This shows that the ratio of the lengths of the sides of the triangle is 1:√3:2, which is the characteristic ratio of a 30-60-90 triangle.

Conclusion

In conclusion, we have determined that the triangle formed by the ground, wall, and ladder is a 30-60-90 triangle. We have used the Pythagorean theorem to find the length of the other leg of the triangle and verified that the ratio of the lengths of the sides is 1:√3:2. This demonstrates the properties of a 30-60-90 triangle and provides a deeper understanding of the mathematics involved in this problem.

Applications of 30-60-90 Triangles

30-60-90 triangles have numerous applications in mathematics, physics, and engineering. Some examples include:

Geometry and Trigonometry: 30-60-90 triangles are used to solve problems involving right-angled triangles, such as finding the length of the sides of a triangle given the lengths of two sides.
Physics: 30-60-90 triangles are used to describe the motion of objects in two-dimensional space, such as the trajectory of a projectile.
Engineering: 30-60-90 triangles are used to design and analyze structures, such as bridges and buildings.

Conclusion

In this article, we have explored the properties of the triangle formed by the ground, wall, and ladder. We have used the Pythagorean theorem to find the length of the other leg of the triangle and verified that the ratio of the lengths of the sides is 1:√3:2. This demonstrates the properties of a 30-60-90 triangle and provides a deeper understanding of the mathematics involved in this problem. The applications of 30-60-90 triangles are numerous and diverse, and this article has provided a brief overview of some of the ways in which they are used in mathematics, physics, and engineering.