Understanding and Calculating the Lengths of Sides in a 3-4-5 Right Triangle

3-4-5 triangle

A 3-4-5 triangle is a right triangle where one of the angles measures 90 degrees

A 3-4-5 triangle is a right triangle where one of the angles measures 90 degrees. The lengths of the sides of the triangle are in the ratio of 3:4:5.

To understand how to find the lengths of the sides, let’s label the triangle. Suppose the side opposite the 90-degree angle is the hypotenuse and has a length of 5 units. The other two sides are the adjacent and opposite sides and have lengths of 3 and 4 units, respectively.

We can use the Pythagorean theorem to verify if this triangle is truly a right triangle. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, for this triangle, we have:
(3^2) + (4^2) = (5^2)
9 + 16 = 25
25 = 25

Since the equation holds true, we confirm that this is indeed a right triangle.

Now, let’s say you are given the length of one side of the triangle, and you need to find the lengths of the other sides.

If you know the length of the hypotenuse (5 units), you can find the lengths of the other two sides using the following ratios:

Adjacent side = (Hypotenuse) * (3/5)
= 5 * (3/5)
= 3 units

Opposite side = (Hypotenuse) * (4/5)
= 5 * (4/5)
= 4 units

Similarly, if you know the length of the adjacent side (3 units), you can find the lengths of the other two sides using the following ratios:

Hypotenuse = (Adjacent side) * (5/3)
= 3 * (5/3)
= 5 units

Opposite side = (Adjacent side) * (4/3)
= 3 * (4/3)
= 4 units

Finally, if you know the length of the opposite side (4 units), you can find the lengths of the other two sides using the following ratios:

Hypotenuse = (Opposite side) * (5/4)
= 4 * (5/4)
= 5 units

Adjacent side = (Opposite side) * (3/4)
= 4 * (3/4)
= 3 units

So, in summary, in a 3-4-5 right triangle, the lengths of the sides can be found using the ratios:
Adjacent side = (Hypotenuse) * (3/5)
Opposite side = (Hypotenuse) * (4/5)

Hypotenuse = (Adjacent side) * (5/3)
Opposite side = (Adjacent side) * (4/3)

Hypotenuse = (Opposite side) * (5/4)
Adjacent side = (Opposite side) * (3/4)

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