In ΔXYZ, y = 6.1 cm, z = 7.5 cm and ∠X=61°. Find the length of x, to the nearest 10th of a centimeter.

To solve this problem, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides of lengths a, b, and c and an angle γ opposite side c, the following equation holds true:

c² = a² + b² - 2ab*cos(γ)

In this case, we know the lengths of sides y and z and the measure of angle X. We can let x = c, y = a, and z = b. Then we substitute these values into the Law of Cosines:

x² = y² + z² - 2yzcos(X) x² = (6.1 cm)² + (7.5 cm)² - 2(6.1 cm)*(7.5 cm)*cos(61°)

Now we just need to calculate the right side of the equation. Remember to convert the angle to radians if your calculator is set to that mode.

x² = 37.21 cm² + 56.25 cm² - 2*(6.1 cm)*(7.5 cm)cos(61°) x² = 93.46 cm² - 91.5 cm²cos(61°)

Calculate the cosine part:

x² = 93.46 cm² - 91.5 cm²*0.4848 x² = 93.46 cm² - 44.37 cm² x² = 49.09 cm²

Finally, we take the square root of both sides to solve for x:

x = sqrt(49.09 cm²) x = 7.0 cm

So, the length of side x is approximately 7.0 cm.

To solve this problem, we can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

The formula for the Law of Sines is: a/sinA = b/sinB = c/sinC

In this case, we know the lengths of sides y and z, and the measure of angle X. We want to find the length of side x.

We can set up the equation as follows:

x/sinX = z/sinZ

We know that sinX = sin61°, z = 7.5 cm, and we want to find x. However, we don't know the value of sinZ.

But we know that in any triangle, the sum of the angles is 180°. So, we can find angle Z by subtracting the known angle X and the right angle (90°) from 180°.

Z = 180° - 90° - 61° = 29°

Now we can substitute the known values into the equation:

x/sin61° = 7.5cm/sin29°

To isolate x, we multiply both sides by sin61°:

x = 7.5cm * sin61° / sin29°

Using a calculator to find the sine values and perform the calculation, we get:

x ≈ 7.5cm * 0.8746197071394 / 0.4848096202463 ≈ 13.5 cm

So, the length of side x, to the nearest tenth of a centimeter, is approximately 13.5 cm.