Question

In questions 16-23 solve \(\Delta \mathrm{XYZ}\) given \(\mathrm{YZ}=15 \mathrm{~cm}, X=39^{\circ}\) and \(Y=75^{\circ}\).

Short answer

Answer: In the triangle XYZ, the side lengths are YZ = 15 cm, XZ ≈ 18.65 cm, and XY ≈ 21.82 cm. The angle measures are X = 39°, Y = 75°, and Z = 66°.

Step-by-step solution

Find the third angle

We know that the sum of the angles in a triangle is always 180°. So, we can find the measure of angle Z by subtracting the measures of angles X and Y from 180°. Z = 180° - X - Y = 180° - 39° - 75° = 66° Now we know all the angles in the triangle: X = 39°, Y = 75°, and Z = 66°.

Use the sine rule to find side XZ

Since we know two angles and one side, we can use the sine rule to find another side length. Let's find the length of side XZ. The sine rule states that \(\frac{\text{side length}}{\sin(\text{opposite angle})}\) is the same for all sides in a triangle. So, for this triangle, we have: \(\frac{\text{YZ}}{\sin(X)}\) = \(\frac{\text{XZ}}{\sin(Y)}\) We know that YZ = 15 cm, X = 39°, and Y = 75°, so we can substitute these values into the sine rule equation: \(\frac{15}{\sin(39°)}\) = \(\frac{\text{XZ}}{\sin(75°)}\) Now, we can solve for XZ: XZ = \(\frac{15\cdot\sin(75°)}{\sin(39°)} \approx 18.65\,\text{cm}\)

Use the sine rule to find side XY

We can use the same sine rule to find the length of side XY: \(\frac{\text{YZ}}{\sin(X)}\) = \(\frac{\text{XY}}{\sin(Z)}\) We know that YZ = 15 cm, X = 39°, and Z = 66°, so we can substitute these values into the sine rule equation: \(\frac{15}{\sin(39°)}\) = \(\frac{\text{XY}}{\sin(66°)}\) Now, we can solve for XY: XY = \(\frac{15\cdot\sin(66°)}{\sin(39°)} \approx 21.82\,\text{cm}\)

Review the complete solution

Now we have found all the side lengths and angle measures for the triangle \(\Delta XYZ\): - Side YZ = 15 cm - Side XZ = 18.65 cm (approximately) - Side XY = 21.82 cm (approximately) - Angle X = 39° - Angle Y = 75° - Angle Z = 66° So, the triangle \(\Delta XYZ\) is completely solved.

Key concepts

Triangle Properties

Triangles are fundamental shapes in geometry defined by three sides and three angles connecting three vertices. Understanding triangle properties is vital because they serve as the building blocks for more complex polygons and geometric principles. Here are a few core properties of triangles that students should know:

• The sum of any triangle's interior angles is always 180 degrees. This essential fact helps in calculating unknown angles when two angles are given.
• Triangles are classified based on their sides (equilateral, isosceles, or scalene) and angles (acute, right, or obtuse). Each class has distinct properties and applications.
• External angles (formed by extending one side of the triangle) are equal to the sum of the two opposite interior angles.

When solving triangle problems, these properties are powerful tools that simplify otherwise complex scenarios. Understanding them firmly is a key step to mastering trigonometry and related topics.

Sine Rule

The sine rule, or law of sines, is an essential formula in trigonometry and is particularly useful for solving problems involving non-right angled triangles. The rule is stated mathematically as follows:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]Here, \(a\), \(b\), and \(c\) are side lengths, while \(A\), \(B\), and \(C\) are the angles opposite these sides, respectively. With this rule, if you know any three of the sides and angles involved, you can find the fourth one.When applied to tasks like those in the original exercise, the sine rule greatly simplifies finding unknown sides (such as sides XZ and XY) when two angles and one side are known. Remember:

• It's crucial to match each side with its corresponding opposite angle.
• Ensure to rearrange the formula based on what you need to find — it can be manipulated to solve for sides or angles, whichever is necessary.

Utilizing the sine rule effectively requires precision in calculation and substitution. Practicing this skill will boost confidence in handling a wide range of trigonometric problems.

Angle Sum Property

The angle sum property might seem simple, but it's a powerful tool in problem-solving. This principle states that the sum of the interior angles of a triangle is always 180 degrees, a rule that holds true regardless of the triangle's type or size.In practice, this property allows for easy calculation of a missing angle when two others are known. As demonstrated in the exercise, with the knowledge of angles X and Y, angle Z can be found by applying:\[ Z = 180^{\circ} - X - Y \]This property is particularly beneficial because:

• It provides a quick way to verify if given angles form a valid triangle.
• It's foundational for working with different geometric shapes, as they are often decomposed into multiple triangles.

Mastering the angle sum property facilitates the learner's ability to solve complex geometry problems efficiently. It appears simple but serves as an indispensable tool across various mathematical disciplines.