In Triangle ATUV, { T = 670 , \text{cm} $}$, { U = 510 , \text{cm} $}$, And { \angle ZV = 159^\circ $}$. Find The Length Of { V $}$, To The Nearest Centimeter.

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Solving for the Length of Side V in Triangle ATUV

In the world of mathematics, particularly in geometry, triangles are a fundamental concept. A triangle is a polygon with three sides and three vertices. In this article, we will focus on solving for the length of side V in triangle ATUV, given the lengths of sides T and U, and the measure of angle ZV.

We are given a triangle ATUV with the following information:

  • The length of side T is 670 cm.
  • The length of side U is 510 cm.
  • The measure of angle ZV is 159°.

Our goal is to find the length of side V to the nearest centimeter.

To solve for the length of side V, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides of length a, b, and c, and angle C opposite side c, the following equation holds:

c² = a² + b² - 2ab * cos(C)

In our case, we can let a = T, b = U, and c = V. We are given the lengths of sides T and U, and the measure of angle ZV. We can plug these values into the equation and solve for V.

Using the Law of Cosines, we can write the equation as follows:

V² = T² + U² - 2TU * cos(ZV)

We are given the lengths of sides T and U, so we can plug these values into the equation:

V² = 670² + 510² - 2(670)(510) * cos(159°)

To calculate the value of V, we need to evaluate the expression inside the parentheses. We can use a calculator to find the value of cos(159°).

cos(159°) ≈ -0.0523

Now we can plug this value back into the equation:

V² = 670² + 510² - 2(670)(510) * (-0.0523)

V² = 448900 + 260100 - 2(670)(510) * (-0.0523)

V² = 709000 + 69551.4

V² = 878551.4

To find the length of side V, we need to take the square root of both sides of the equation:

V = √(878551.4)

V ≈ 941.9 cm

Since we are asked to find the length of side V to the nearest centimeter, we can round our answer to the nearest whole number:

V ≈ 942 cm

In this article, we used the Law of Cosines to solve for the length of side V in triangle ATUV. We were given the lengths of sides T and U, and the measure of angle ZV. We plugged these values into the equation and solved for V. Our final answer is V ≈ 942 cm.

  • When using the Law of Cosines, make sure to plug in the correct values for the sides and angles.
  • Use a calculator to evaluate expressions and find the value of cos(θ).
  • Round your answer to the nearest whole number when asked to do so.

By following these tips and tricks, you can become more confident in your ability to solve problems involving the Law of Cosines.
Frequently Asked Questions (FAQs) about Solving for the Length of Side V in Triangle ATUV

A: The Law of Cosines is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used to solve for the length of a side of a triangle when we know the lengths of the other two sides and the measure of the angle between them. In the case of triangle ATUV, we can use the Law of Cosines to solve for the length of side V by plugging in the given values for the lengths of sides T and U, and the measure of angle ZV.

A: To use the Law of Cosines to solve for the length of side V, follow these steps:

  1. Write down the equation for the Law of Cosines: c² = a² + b² - 2ab * cos(C)
  2. Plug in the given values for the lengths of sides a and b, and the measure of angle C.
  3. Evaluate the expression inside the parentheses using a calculator.
  4. Plug the value of cos(C) back into the equation.
  5. Simplify the equation and solve for c².
  6. Take the square root of both sides of the equation to find the length of side c.

A: Some common mistakes to avoid when using the Law of Cosines to solve for the length of side V include:

  • Plugging in the wrong values for the sides and angles.
  • Not evaluating the expression inside the parentheses correctly.
  • Not simplifying the equation correctly.
  • Not taking the square root of both sides of the equation.

A: Yes, the Law of Cosines can be used to solve for the length of side V in any type of triangle, as long as we know the lengths of the other two sides and the measure of the angle between them.

A: The Law of Cosines has many real-world applications, including:

  • Navigation: The Law of Cosines can be used to calculate the distance between two points on the surface of the Earth.
  • Surveying: The Law of Cosines can be used to calculate the length of a side of a triangle formed by three survey points.
  • Physics: The Law of Cosines can be used to calculate the length of a side of a triangle formed by three objects in motion.

A: You can practice using the Law of Cosines to solve for the length of side V by working through example problems and exercises. You can also use online resources and calculators to help you practice and check your work.

A: Some additional tips and tricks for using the Law of Cosines to solve for the length of side V include:

  • Make sure to plug in the correct values for the sides and angles.
  • Use a calculator to evaluate expressions and find the value of cos(θ).
  • Simplify the equation correctly before solving for the length of side V.
  • Check your work by plugging the value of the length of side V back into the equation to make sure it is true.

By following these tips and tricks, you can become more confident in your ability to use the Law of Cosines to solve for the length of side V in triangle ATUV.