Understanding the Properties of a 3-4-5 Triangle: Side Ratios, Right Angles, and Trigonometry

3-4-5 triangle

A 3-4-5 triangle is a right triangle where the lengths of its sides are in the ratio 3:4:5

A 3-4-5 triangle is a right triangle where the lengths of its sides are in the ratio 3:4:5. In other words, the lengths of the sides are such that if you multiply each side by a constant factor, you will always get a 3-4-5 triangle.

To understand the properties of a 3-4-5 triangle, let’s first define the sides. The shortest side is known as the “opposite side” or “height” and has a length of 3. The middle side is known as the “adjacent side” or “base” and has a length of 4. The longest side is known as the “hypotenuse” and has a length of 5.

The most important property of a 3-4-5 triangle is that it is a right triangle, meaning it has one angle that measures 90 degrees. This is because of the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In a 3-4-5 triangle, we can use the Pythagorean theorem to verify that it is indeed a right triangle:

(3^2) + (4^2) = 9 + 16 = 25

Since 25 is equal to the square of 5, it satisfies the Pythagorean theorem, confirming that the triangle is right-angled.

The angles of a 3-4-5 triangle can also be determined using trigonometry. The smallest angle is opposite the smallest side (3), and the largest angle is opposite the largest side (5). By using inverse trigonometric functions, we can find the measures of these angles.

The trigonometric functions sine, cosine, and tangent are commonly used in right triangles. In this case, we can calculate the sine and cosine functions for the angles as follows:

sin(smallest angle) = opposite/hypotenuse = 3/5
cos(smallest angle) = adjacent/hypotenuse = 4/5

We can find the values of these angles from the inverse sine and inverse cosine functions on a calculator:

smallest angle ≈ arcsin(3/5) ≈ 36.87 degrees
largest angle ≈ arccos(4/5) ≈ 53.13 degrees

The remaining angle of the triangle can be found by subtracting the sum of the other two angles from 180 degrees:

remaining angle = 180 – smallest angle – largest angle
= 180 – 36.87 – 53.13
≈ 90 degrees

In summary, a 3-4-5 triangle is a right triangle with side lengths of 3, 4, and 5. It satisfies the Pythagorean theorem and has angles measuring approximately 36.87 degrees, 53.13 degrees, and 90 degrees.

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