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Last modified on August 3rd, 2023

As we know, special right triangles can have different dimensions, but among them, the most common is the 3-4-5 right triangle. Here, we will learn about 3-4-5 right triangles and how to solve problems involving them.

A 3-4-5 triangle is a special right triangle whose side lengths are in the ratio of 3: 4: 5. It is thus a right triangle with sides in the ratio of integer lengths (whole numbers) called Pythagorean triples. Since all its side lengths are different from the other; it is also called a scalene-right triangle.

Being a right triangle, the** **Pythagoras formula a^{2} + b^{2} = c^{2}, where a = side 1, b = side 2, and c = hypotenuse is also applicable in a 3-4-5 triangle.

In a** **3-4-5 triangle,

a^{2} + b^{2} = c^{2}, here a = 3 units, b = 4 units

⇒ 3^{2} + 4^{2} = c^{2}

⇒ 25 = c^{2}

⇒ c = 5 units

Hence, it works.

**Other Pythagorean Triples**

3-4-5 triangle does not mean that the ratios are always exactly 3: 4: 5. But, it can be any factor of numbers, keeping the basic ratio of the three sides the same. Few other examples of 3-4-5 triangles are:

- 6-8-10
- 9-12-15
- 12-16-20
- 15-20-25

- All three sides are unequal, having a ratio of 3: 4: 5 (3x: 4x: 5x for Side 1: side 2: hypotenuse).
- All three internal angles are unequal measuring 36.87°, 53.13°, and 90°

Solving a 3-4-5 right triangle involves finding the missing side lengths of the triangle. The ratio of 3: 4: 5 allows us to calculate the unknown lengths without using the Pythagorean Theorem or trigonometric functions.

**Find the length of the unknown side of a right triangle in which hypotenuse measures 35 cm and one of the other side measures 28 cm.**

Solution:

We first have to prove whether the given triangle is a 3-4-5 triangle, for which the test ratio should be 3:4:5.

?: 28: 35 = ?: 4(7): 5(7)

Here, factor x = 7

Thus, the given triangle is a 3-4-5 right triangle

Hence, the length of the unknown side is:

3x = 3 x 7 = 21 cm

**Find the value of y in the given triangle. Assume that the triangle is a 3-4-5 right triangle.**

Solution:

We first have to prove whether the given triangle is a 3-4-5 triangle, for which the test ratio should be 3:4:5

?: 100: 125 = ?: 4(25): 5(25)

Here, factor x = 25

Thus, the given triangle is a 3-4-5 right triangle

Hence, the length of the unknown side (y) is:

y = 3x = 3 x 25 = 75 cm

**Calculate the length of the diagonal of a right triangle with side lengths of 9mm and 12mm.**

Solution:

We first have to prove whether the given triangle is a 3-4-5 triangle, for which the test ratio should be 3:4:5

9: 12: ? = 3(3): 4(3): ?

Here, factor x = 3

Thus, the given triangle is a 3-4-5 right triangle,

Substituting the value of x, we get

Diagonal = hypotenuse = 5x = 5 x 3 =15mm

Hence, the length of the diagonal is 15mm.

**The longest edge and bottom edge of triangular-shaped sail of a boat is 20m and 16m respectively. Find the height of the sail?**

Solution:

Since, we first need to check whether the triangular sail makes a 3-4-5 right triangle with the bottom, we will find its test ratio.

⇒ ?: 16: 20 = ? : 4(4): 5(4)

Here, factor x = 4

Thus, the given triangle is a 3-4-5 right triangle

Substituting the value of x, we get Height = 3x = 3 x 4 = 12m

Hence, the height of the sail is 12m.

Last modified on August 3rd, 2023