The triangle inequality theorem states that, in a triangle, the sum of lengths of any two sides is greater than the length of the third side. Suppose a, b and c are the lengths of the sides of a triangle, then, the sum of lengths of a and b is greater than the length c. Similarly, b + c > a, and a+ c > b. If, in any case, the given side lengths are not able to satisfy these conditions, it means it is not possible to draw a triangle with those measurements. Show
What is Triangle Inequality Theorem?The triangle inequality theorem states, "The sum of any two sides of a triangle is greater than its third side." This theorem helps us to identify whether it is possible to draw a triangle with the given measurements or not without actually doing the construction. Let's understand this with the help of an example. Triangle ABC has side lengths of 6 units, 8 units, and 12 units. Here, AB = 6 units, BC = 8 units and CA = 12 units.
Thus, lengths of all the sides satisfy the triangle inequality theorem. In this, not only one, but all 3 cases should satisfy the triangle inequality theorem. Let's take another example. Let's check whether a triangle with sides lengths 5 units, 3 units, and 10 units satisfy the triangle inequality theorem or not. Here,
We can see that two cases are satisfying the triangle inequality theorem but one case is not satisfying. This means the triangle with these side lengths does not exist. All three sides should satisfy the triangle inequality theorem. Triangle Inequality Theorem FormulaBefore understanding the formula, first, we need to understand the proof of the triangle inequality theorem. Consider a triangle ABC as shown below. Let us extend side AB to the point D such that AC = AD and △BDC will form a right angled triangle at angle C. Applying angle sum property in △BDC, we get, ∠BDC + ∠CBD + ∠BCD = 180° ∠BDC + ∠CBD + 90° = 180° ∠BDC + ∠CBD = 90° This implies, ∠BCD > ∠BDC. As the side opposite to the greater angle is longer, we have BD > BC. This implies: BD > BC AB + AD > BC AB + AC > BC Hence proved. Similarly, we can prove that AC + BC > AB and AB + BC > AC. So, the triangle inequality theorem formula is,
Related Articles on Triangle Inequality TheoremCheck the following articles to learn more about the triangle inequality theorem.
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FAQs on Triangle Inequality TheoremThe triangle inequality theorem states that the sum of any two sides of a triangle is greater than the third side, and if the sum of any two sides of a triangle is not greater than the third side it means the triangle does not exist. Why is Triangle Inequality Theorem Important?The triangle inequality theorem is important to find out whether the triangle with the given three measurements exists or not. As the theorem states that sum of any two sides should be greater than the measurement of the third side. For example, the triangle with sides 3 units, 4 units, and 9 units does not exist as it does not satisfy the triangle inequality theorem.
Thus, by using the triangle inequality theorem we can say that the given measurements do not form a triangle. How is the Triangle Inequality Theorem used in Real Life?One example of the application of the triangle inequality theorem in real life is by Engineers. Civil engineers use the triangle inequality theorem in real life. Since their work is related to surveying, transportation, and urban planning. With the help of the triangle inequality theorem, they calculate the unknown lengths and estimate the remaining dimension. Does the Triangle Inequality Theorem Apply to all Triangles?Yes, the triangle inequality theorem applies to all triangles. Any side of a triangle must be shorter than the sum of the other two sides. If a side is greater than or equal to the sum of the other two sides, then it is not a triangle. What is an example of the Triangle Inequality Theorem?Following is the example of the triangle inequality theorem. Triangle with side lengths 5, 7, and 9 units exists, as lengths of all sides satisfy the theorem.
How do you write a Triangle Inequality Theorem?Suppose ABC is a triangle. We will write the triangle inequality theorem in this form:
Hannah B. the sum of the lengths of any 2 sides of a triangle must be greater than the third side. one side is 17cm and second side is 1 cm less than twice the third side. what are the possible lengths for the second and third sides? 1 Expert Answer
Jordan K. answered • 10/20/15 Nationally Certified Math Teacher (grades 6 through 12)
Let's begin by identifying the lengths of the three sides of the triangle: length of side 2 = 2x - 1 (1 less than twice side 3) Now let's apply the Triangle Inequality Theorem to this triangle: side 1 + side 2 > side 3: x > -16 (reject negative measurement) 2x - 1 > -33 (reject negative measurement) Thus, we have our answers based on the value of x: 6 < x (length of side 3) < 18 11 < 2x - 1 (length of side 2) < 35 Thanks for submitting this problem and glad to help. |