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Chapter 10: Problem 23

In questions 16-23 solve \(\Delta \mathrm{XYZ}\) given \(\mathrm{XY}=100 \mathrm{~cm}, \mathrm{XZ}=73 \mathrm{~cm}\) and \(Y=50^{\circ}\).

Short Answer

Expert verified
Answer: In triangle XYZ, side YZ ≈ 97.82 cm, angle X ≈ 54.25°, and angle Z ≈ 75.75°.

Step by step solution

01

Identify the given information

We know that side XY = 100 cm, side XZ = 73 cm, and angle Y = 50°.
02

Use the Law of Cosines to find side YZ

The Law of Cosines is given by the formula: c^2 = a^2 + b^2 - 2ab * cos(C), where a, b, c are sides of the triangle and C is the angle opposite to side c. In this case, we have side XY (a) = 100 cm, side XZ (b) = 73 cm, and angle Y (C) = 50°. We need to find side YZ (c). Plug the given values into the formula: c^2 = (100)^2 + (73)^2 - 2(100)(73) * cos(50°) Using a calculator, we get: c^2 ≈ 9569.951 Now, take the square root of both sides: c ≈ 97.82 cm (approximately) So, side YZ = 97.82 cm.
03

Use the Law of Sines to find angle X

The Law of Sines is given by the formula: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are sides of the triangle and A, B, C are the respective angles. We have side YZ (c) = 97.82 cm and angle Y (C) = 50°. Let's find angle X (A) using side XY (a) = 100 cm. sin(A)/100 = sin(50°)/97.82 Now, solve for angle A (X) using the inverse sine function: A = sin^(-1)((100 * sin(50°))/97.82) ≈ 54.25° (approximately) So, angle X = 54.25°.
04

Find angle Z

Since the sum of angles in a triangle is always 180°, we can find angle Z: Z =180° - (angle X + angle Y) = 180° - (54.25° + 50°) = 75.75° (approximately) So, angle Z = 75.75°.
05

State the final solution

The solved triangle XYZ has the following measures: Side YZ ≈ 97.82 cm, angle X ≈ 54.25°, and angle Z ≈ 75.75°.

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