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Lesson Contents

What is a 3 4 5 Triangle?

A 3 4 5 triangle is an SSS right triangle (meaning we know the three side lengths).
If we know two of the side lengths and they are congruent with the 3 4 5 ratio, we can easily determine the third side length by using the ratio.
The other common SSS special right triangle is the 5 12 13 triangle.

We call it the 3 4 5 “ratio” because the side lengths do not need to be exactly 3, 4, and 5, but rather can be any common factor of these numbers. For example, a right triangle with side lengths of 6, 8, and 10 is considered a 3 4 5 triangle. Its side lengths are a common factor of 2 of the 3 4 5 ratio.

A 3 4 5 triangle is classified as a scalene triangle since all three sides lengths and internal angles are different.



              A triangle with side lengths in the 3 4 5 ratio.

List of 3 4 5 Triangles

The following are some of the most common sets of 3 4 5 triangle side lengths. We do not need to memorize this list as long as we are aware that the ratio can be multiplied by a common factor to produce any of these sets of side lengths.
-3 4 5
-6 8 10
-9 12 15
-12 16 20
-15 20 25

3 4 5 Triangle Angles

3 4 5 triangle angles

How to Construct a 3 4 5 Triangle

We can construct a 3 4 5 triangle by starting with a two lines that meet at a right angle. Make the vertical line about 3/4 as long as the horizontal line. Then, connect the ends of these two lines with a straight line.

The connecting line makes our hypotenuse, which will end up being about 5/4 as long as the horizontal side, and about 5/3 as long as the vertical side. Label the short leg 3, the long leg 4, and the hypotenuse 5.

Equations for Area and Perimeter

The equation for area of a 3 4 5 triangle is given as:
Area =
1/2bh


This becomes area =
1/2(3)(4) = 6 when it is a true 3 4 5 triangle.
If the triangle is scaled from the ratio by a common factor, we can multiply 6 by that common factor to get the area.

The equation for perimeter is given as:
Perimeter = a + b + c = 3 + 4 + 5 = 12
for a true 3 4 5 triangle.
If the triangle is scaled from the ratio by a common factor, we can multiply 12 by that common factor to get the perimeter.

Proof of the 3 4 5 Triangle

Let’s take a look at the math that proves the existence of the 3 4 5 ratio. We are going to use the standard side lengths of 3 and 4 to look for the 3rd side length using the Pythagorean theorem.
1.) Remember that a2
+ b2
= c2.
2.) 32 + 42 = 9 + 16 = 25
3.) 52 = 25, so the 3 4 5 right triangle ratio is satisfied.

Let’s prove it again with a different example. We are going to multiply the numbers of the ratio by a common factor of 2.
1.) 62 + 82 = 36 + 64 = 100
2.) 102 = 100, so the 3 4 5 right triangle ratio is satisfied.
3.) This proves the ratio holds true even when scaled by a common factor.

Baca :   Diketahui Dalam Koordinat Kartesius Terdapat Titik P

3 4 5 Triangle Example Problem

Which of the following triangles are 3 4 5 right triangles?
Triangle 1: 8, 8, 25
Triangle 2: 9, 12, 15
Triangle 3: 23, 27, 31
Triangle 4: 12, 16, 20
Triangle 5: 6, 8, 10

Solution:
1.) Triangle 2 is the 3 4 5 ratio multiplied by a common factor of 3.
2.) Triangle 4 is the 3 4 5 ratio multiplied by a common factor of 4.
3.) Triangle 5 is the 3 4 5 ratio multiplied by a common factor of 2.
4.) Therefore,
triangles 2, 4, and 5 are 3 4 5 right triangles.

3 5 3 10 4 15

Sumber: https://www.voovers.com/geometry/3-4-5-triangle/

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