![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/03\/image.png\")
<\/figure>\n\n\n\nThe most basic 3-4-5 triangle is the right triangle with the following side lengths: 3 inches, 4 inches, and 5 inches. All right triangles will satisfy the Pythagorean theorem, so the 3-4-5 right triangle is no exception. The Pythagorean theorem states that the sum of the sides\u2019 squares must be equal to the square of the right triangle\u2019s hypotenuse. <\/p>\n\n\n\n
[katex]\\begin{aligned}a^2 + b^2 &= c^2\\\\ (3)^2 + (4)^2 &= (5)^2\\\\ 9 +16&= 25\\\\25 &\\overset{\\checkmark}{=} 25 \\end{aligned}[\/katex]<\/p>\n\n\n\n
As expected, the right triangle shown above satisfies the Pythagorean theorem. This is true for all 3-4-5 right triangles. As long as the sides of a scalene right triangle reduce to a ratio of 3:4:5, the triangle is a 3-4-5 right triangle. Before generalizing the process of finding these triangles, take a look at the right triangle shown below first.<\/p>\n\n\n\n Observe the side lengths of the right triangle: 9 cm, 12 cm, and 15 cm. Write down the ratio of the sides then simplify to see if it is also a 3-4-5 right triangle. <\/p>\n\n\n\n [katex]9 : 12: 15 = 3: 4 : 5[\/katex]<\/p>\n\n\n\n By simplifying the ratio of the triangle\u2019s side, we can confirm that the triangle is indeed a 3-4-5 right triangle. As expected, this 3-4-5 right triangle also satisfies the Pythagorean theorem. <\/p>\n\n\n\n [katex]\\begin{aligned}a^2 + b^2 &= c^2\\\\ (9)^2 + (12)^2 &= (15)^2\\\\ 81 +144&= 225\\\\225 &\\overset{\\checkmark}{=} 225 \\end{aligned}[\/katex]<\/p>\n\n\n\n In fact, all 3-4-5 right triangles showcase side lengths that are Pythagorean triples. This means that knowing why and how to identify Pythagorean triples is helpful when working with 3-4-5 right triangles.<\/p>\n\n\n\n The side lengths of a 3-4-5 right triangle are always Pythagorean triples– these are set of integers that satisfy the Pythagorean theorem. Similar to other Pythagorean triples, the multiples of the set, [katex]\\{3, 4, 5\\}[\/katex], will also be side lengths of 3-4-5 right triangles. This means that when a right triangle has any of these Pythagorean triples, it is going to be a 3-4-5 right triangle. <\/p>\n\n\n\n This table can go on endlessly since any factor can be multiplied to each of the values from {3, 4, 5} to form a new set of a Pythagorean triple. Suppose that n<\/em> is a positive factor, the Pythagorean triple, [katex]\\{3n, 4n, 5n\\}[\/katex] represents the sides of a 3-4-5 right triangle. <\/p>\n\n\n\n The triangle shown above is the general form of the 3-4-5 right triangle. As with all right triangles, the general form satisfies the Pythagorean theorem as well. Substitute these side lengths in terms of n<\/em> into the Pythagorean theorem\u2019s equation.<\/p>\n\n\n\n [katex]\\begin{aligned}a^2 + b^2 &= c^2\\\\ (3n)^2 + (4n)^2 &= (5n)^2\\\\ 9n^2 +16n^2 &= 25n^2\\\\25n^2 &\\overset{\\checkmark}{=} 25n^2 \\end{aligned}[\/katex]<\/p>\n\n\n\n This confirms that the side lengths of any 3-4-5 right triangle will always be a Pythagorean theorem. In addition, we can find different side lengths for a 3-4-5 right triangle by multiplying the Pythagorean triple, {3, 4, 5}, by a positive integer.<\/p>\n\n\n\n Apply what you\u2019ve learned in this section to identify the 3-4-5 right triangles among the five right triangles shown below. <\/p>\n\n\n\n To confirm that a given right triangle is a 3-4-5 right triangle, find the ratios of the side lengths then see if they reduce to 3:4:5. If the ratio reduces to 3:4:5, we can confirm that the triangle is indeed a 3-4-5 right triangle. The table below summarizes the side length ratios of the five triangles.<\/p>\n\n\n\n There are only two triangles with side length ratios not equal to 3:4:5 \u2013 these are triangles B and E. This means that the remaining triangles are 3-4-5 right triangles. Hence, triangles A, C, and D are all 3-4-5 right triangles. Now that we\u2019ve established the rules for identifying 3-4-5 right triangles, it\u2019s time to expand our understanding and learn how to apply them when solving unknown measures and problems.<\/p>\n\n\n\n Utilize the common ratio that 3-4-5 right triangles share when solving problems involving them. When looking for an unknown measure of a 3-4-5 right triangle, identify the side that corresponds with the ratio\u2019s component. Simplify the process of solving problems by utilizing the Pythagorean triple, {3, 4,5}, and its corresponding multiples.<\/p>\n\n\n\n In this section, we\u2019ll cover some examples showing how helpful 3-4-5 right triangles can be when solving problems. When dealing with 3-4-5 right triangles, knowing their properties will make a difference. Here are the properties of 3-4-5 right triangles that will come in handy later on:<\/p>\n\n\n\n Utilizing these properties simplifies the steps for solving unknown measures of a right triangle. Learn how without using Pythagorean\u2019s theorem, it\u2019s still possible to solve and work with these special right triangles.<\/p>\n\n\n\n A right triangle has a leg that measures 48 cm and a hypotenuse that is 80 cm long. What is the length of the right triangle\u2019s remaining leg?<\/p>\n\n\n\n When working with problems like this, the first thing to do is to confirm that we\u2019re working with a 3-4-5 right triangle. To do so, find the ratio of the two remaining sides to see if they simply to two components of 3:4:5 with 80 cm corresponding to the third ratio component.<\/p>\n\n\n\n Let h<\/em> be the unknown leg of the right triangle, so the ratio of the right triangle\u2019s side lengths is h: <\/em>48: 80. When simplified, the ratio of the three sides becomes h\/16 <\/em>: 3: 5. This crosses off our condition \u2013 the simplified ratio must have at least two components of 3: 4: 5. <\/p>\n\n\n\n [katex]h: 48: 80 = \\dfrac{h}{16}: 3: 5[\/katex]<\/p>\n\n\n\n Rearranging the ratio of the right triangle\u2019s side lengths, we have 3: h\/16<\/em> : 5. This means that the unknown leg represents the 3-4-5 right triangle\u2019s side that is a multiple of 4. To find the value of h<\/em>, simply equate h\/16<\/em> to 4.<\/p>\n\n\n\n [katex]\\begin{aligned}\\dfrac{h}{16} &= 4\\\\h &= 4(16)\\\\ h&= 64\\end{aligned}[\/katex]<\/p>\n\n\n\n This means that the unknown leg\u2019s length is 64 cm. We can confirm this by seeing if the side lengths reduce to 3:4:5.<\/p>\n\n\n\n [katex]\\begin{aligned}48: 64:80 = 3: 4: 5\\end{aligned}[\/katex]<\/p>\n\n\n\n A rectangle is formed by two right triangles as shown below. What is the diagonal of the rectangle?<\/p>\n\n\n\n The diagonal of the rectangle is also the hypotenuse of each of the right triangles. Now, let h<\/em> represent the hypotenuse of the right triangle. Observe the ratio of the right triangle\u2019s sides first to see if we can apply the properties of 3-4-5 triangles for this problem.<\/p>\n\n\n\n [katex]27: 36: h = 3: 4: h[\/katex]<\/p>\n\n\n\n Since the legs of the right triangle have side lengths with a ratio of 3:4, we\u2019re working with a 3-4-5 right triangle. The common factor shared between the legs is 9, so h<\/em> must also share this as a common factor along with 5 as another factor. <\/p>\n\n\n\n [katex]\\begin{aligned}h &= 9(5)\\\\&= 45\\end{aligned}[\/katex]<\/p>\n\n\n\n This means that the value of h<\/em> is equal to 45 inches. We can also confirm this by simplifying 27:36:45 \u2013 it returns a ratio of 3:4:5. Hence, the diagonal of the rectangle is also 45 inches long.<\/p>\n","protected":false},"excerpt":{"rendered":" In this article, we will cover the properties that define 3, 4, 5 triangles and learn how to identify these triangles by inspecting their sides and angles.<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-460","post","type-post","status-publish","format-standard","hentry","category-geometry"],"_links":{"self":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/460","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/comments?post=460"}],"version-history":[{"count":2,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/460\/revisions"}],"predecessor-version":[{"id":470,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/posts\/460\/revisions\/470"}],"wp:attachment":[{"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/media?parent=460"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/categories?post=460"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/visualfractions.com\/blog\/wp-json\/wp\/v2\/tags?post=460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/03\/image-1.png\")
<\/figure>\n\n\n\nPythagorean Triples and 3 4 5 Right Triangles<\/strong><\/h2>\n\n\n\n
Pythagorean Triples<\/strong><\/td><\/tr> {3, 4, 5}<\/td> {6, 8, 10}<\/td> {9, 12, 15}<\/td><\/tr> {12, 16, 20}<\/td> {15, 20, 25}<\/td> {18, 24, 30}<\/td><\/tr> {21, 28, 35}<\/td> {24, 32, 40}<\/td> {27, 36, 45}<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n ![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/03\/image-2.png\")
<\/figure>\n\n\n\nProblem 1<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/03\/image-3.png\")
<\/figure>\n\n\n\nTriangle A<\/strong><\/td> [katex]6:8:10 = 3:4:5[\/katex]<\/td><\/tr> Triangle B<\/strong><\/td> [katex]10:24:26 = 5:12:13[\/katex]<\/td><\/tr> Triangle C<\/strong><\/td> [katex]30: 40:50 = 3:4:5[\/katex]<\/td><\/tr> Triangle D<\/strong><\/td> [katex]18 :24:30 = 3: 4 :5[\/katex]<\/td><\/tr> Triangle E<\/strong><\/td> [katex]10:30:34 = 5:15:17[\/katex]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n How To Solve 3 4 5 Right Triangle Problems?<\/h2>\n\n\n\n
Problem 2<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/03\/image-4.png\")
<\/figure>\n\n\n\nProblem 3<\/h3>\n\n\n\n
![\"\"](\"https:\/\/visualfractions.com\/blog\/wp-content\/uploads\/2022\/03\/image-5.png\")
<\/figure>\n\n\n\n