Unit 5 Relationships In Triangles Homework 6 Triangle Inequalities UPD

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Unit 5 Relationships in Triangles Homework 6 Triangle Inequalities

Triangles are one of the most basic and important shapes in geometry. They have many properties and relationships that can help us solve problems and understand the world around us. One of these properties is the triangle inequality, which states that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

In this article, we will explore the triangle inequality and how it relates to other concepts in unit 5 relationships in triangles. We will also look at some examples and exercises from homework 6 that will help you practice and master this topic.

What is the Triangle Inequality?

The triangle inequality is a simple but powerful rule that applies to any triangle. It says that if you add up the lengths of any two sides of a triangle, the result will always be bigger than the length of the third side. For example, if a triangle has sides of 3 cm, 4 cm, and 5 cm, then 3 + 4 = 7 cm, which is greater than 5 cm. This means that the triangle inequality holds for this triangle.

The triangle inequality can also be written as an inequality involving three variables: a, b, and c. If a, b, and c are the lengths of the sides of a triangle, then the triangle inequality can be expressed as:

a + b > c

b + c > a

c + a > b

These three inequalities are equivalent and can be derived from each other by switching the variables around. They all mean the same thing: the sum of any two sides of a triangle is greater than the third side.

Why is the Triangle Inequality Important?

The triangle inequality is important because it tells us something about the shape and size of a triangle. It also helps us determine whether three given lengths can form a triangle or not. For example, if we have three lengths of 2 cm, 3 cm, and 6 cm, can we make a triangle with them? To answer this question, we can use the triangle inequality and check if any of the sums of two lengths are greater than the third length. In this case, we find that:

2 + 3 = 5 cm < 6 cm

3 + 6 = 9 cm > 2 cm

6 + 2 = 8 cm > 3 cm

Since one of the sums is not greater than the third length (2 + 3 < 6), we conclude that these three lengths cannot form a triangle. This means that not every combination of three lengths can make a triangle. There are some restrictions on how long or short the sides can be.

How Does the Triangle Inequality Relate to Other Concepts in Unit 5 Relationships in Triangles?

The triangle inequality is related to many other concepts in unit 5 relationships in triangles. Here are some examples:

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle. This theorem can be proved using the triangle inequality and some algebra.

The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two remote interior angles. This theorem can be proved using the triangle inequality and some geometry.

The Inequality Theorems for Two Triangles state that if two triangles are similar or congruent, then corresponding sides and angles have proportional or equal measures. These theorems can be proved using the triangle inequality and some ratios.

The Inequalities in One Triangle state that in any triangle, the longest side is opposite to the largest angle, and vice versa. These inequalities can be proved using the triangle inequality and some logic.

What are Some Examples and Exercises from Homework 6?

To help you practice and master the topic of triangle inequalities, here are some examples and exercises from homework 6:

Example: Given that ABC is a triangle with AB = 12 cm, BC = 15 cm, and CA = 9 cm, find ACB.

Solution: To find ACB, we can use the law of cosines, which states that for any triangle ABC with sides a, b, and c opposite to angles A, B, and C respectively:

c^2 = a^2 + b^2 - 2ab cos C

Plugging in the given values, we get:

(9)^2 = (12)^2 + (15)^2 - 2(12)(15) cos C

Solving for cos C, we get:

cos C = -0.4

Taking the inverse cosine of both sides, we get:

C = arccos(-0.4)

C â‰ˆ 113.58Â°

Therefore, ACB â‰ˆ 113.58Â°.

Exercise: Given that XYZ is a right triangle with XY = 8 cm and XZ = 10 cm, find YZ.

Hints:

You can use the Pythagorean theorem to find YZ.

The Pythagorean theorem states that for any right triangle XYZ with hypotenuse XZ:

XZ^2 = XY^2 + YZ^2

Rearrange this equation to solve for YZ.

Simplify your answer by taking its square root.

Conclusion

In this article, we have learned about the triangle inequality and how it applies to any triangle. We have also seen how it relates to other concepts in unit 5 relationships in triangles, such as the Hinge Theorem, the Exterior Angle Theorem, the Inequality Theorems for Two Triangles, and the Inequalities in One Triangle. We have also looked at some examples and exercises from homework 6 that will help you practice and master this topic.

Triangle inequalities are an important and useful tool in geometry. They can help us determine whether three lengths can form a triangle or not, and they can also help us compare and contrast different triangles based on their sides and angles. By understanding and applying the triangle inequality, we can solve many problems and explore many relationships in triangles.

How to Apply the Triangle Inequality to Solve Problems?

One of the most common applications of the triangle inequality is to solve problems involving the lengths of sides or angles of triangles. Here are some steps to follow when using the triangle inequality to solve problems:

Identify the given information and the unknown quantity.

Draw a diagram of the situation and label the sides and angles accordingly.

Use the triangle inequality to write an inequality involving the given and unknown quantities.

Solve the inequality for the unknown quantity by using algebraic techniques.

Check your answer by plugging it back into the inequality and making sure it is true.

Let's look at an example of how to apply these steps:

Example:

A triangle has sides of 7 cm and 10 cm. What is the range of possible lengths for the third side?

Solution:

We are given two sides of a triangle and we need to find the range of possible lengths for the third side. Let's call the third side x.

We can draw a diagram of the triangle and label the sides as follows:

/\

/ \

/ \

/ \

/________\

7 x 10

We can use the triangle inequality to write three inequalities involving x:

7 + 10 > x

10 + x > 7

x + 7 > 10

We can simplify these inequalities by subtracting or adding terms on both sides:

x < 17

x > -3

x > 3

We can combine these inequalities into one compound inequality:

3 < x < 17

This means that x must be greater than 3 and less than 17. This is the range of possible lengths for the third side.

We can check our answer by plugging in some values for x and making sure they satisfy the triangle inequality. For example, if x = 5, then we have:

7 + 10 = 17 > 5

10 + 5 = 15 > 7

5 + 7 = 12 > 10

All three inequalities are true, so x = 5 is a valid length for the third side. Similarly, we can check other values for x within the range and make sure they work.

What are Some Common Mistakes to Avoid When Using the Triangle Inequality?

The triangle inequality is a simple but powerful rule that can help us solve many problems involving triangles. However, there are some common mistakes that we should avoid when using the triangle inequality. Here are some examples of what not to do:

Do not assume that equality holds in the triangle inequality. The triangle inequality states that the sum of any two sides of a triangle is greater than the third side, not equal to it. Equality only holds when the triangle is degenerate, meaning that it has zero area and all three points are collinear. For example, if we have three lengths of 4 cm, 5 cm, and 9 cm, then we cannot form a triangle with them because 4 + 5 = 9, which violates the triangle inequality. However, we can form a line segment with these lengths by placing them end to end.

Do not confuse the order of subtraction in the triangle inequality. The triangle inequality states that the sum of any two sides of a triangle is greater than the third side, not less than it. This means that we cannot subtract any two sides from each other and compare them to the third side. For example, if we have a triangle with sides of 6 cm, 8 cm, and 10 cm, then we cannot say that 6 - 8 < 10 or that 8 - 6 < 10. These statements are false and do not follow from the triangle inequality. However, we can say that 10 - 6 < 8 or that 10 - 8 < 6. These statements are true and follow from rearranging the triangle inequality.

Do not forget to check all three inequalities in the triangle inequality. The triangle inequality states that the sum of any two sides of a triangle is greater than the third side for all three combinations of sides. This means that we have to check all three inequalities when determining whether three lengths can form a triangle or not. For example, if we have three lengths of 2 cm, 4 cm, and 7 cm, then we cannot form a triangle with them because one of the inequalities does not hold: 2 + 4 = 6 < 7. However, if we only check two of the inequalities, we might mistakenly think that they can form a triangle: 2 + 7 = 9 > 4 and

4 + 7 =11 >2.

Conclusion

In these paragraphs, we have learned how to apply the triangle inequality to solve problems involving the lengths of sides or angles of triangles. We have also seen some common mistakes to avoid when using the triangle inequality. The triangle inequality is a useful and important rule that can help us understand and compare different triangles based on their properties and relationships. By using the triangle inequality correctly and carefully, we can solve many problems and explore many aspects of triangles. 4aad9cdaf3