Math Problem Statement

The image contains two triangle problems: one is a right-angled triangle with known side lengths, and the other is a non-right-angled triangle with a given angle and two sides.

Solution

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Problem Statement

The image shows two triangles. The left triangle is a right-angled triangle with sides labeled as follows:

  • The side opposite the right angle is labeled c=17c = 17 cm.
  • One leg is labeled a=15a = 15 cm.
  • The other leg is labeled bb.

The right triangle is a non-right-angled triangle labeled as follows:

  • One side is labeled c=10c = 10 cm.
  • Another side is labeled b=8b = 8 cm.
  • The angle between these sides is labeled α=60\alpha = 60^\circ.

Task

  1. For the left triangle, we need to find the length of the unknown side bb.
  2. For the right triangle, we need to find the length of the third side using the given angle.

Step-by-Step Solution

  1. For the Left Triangle (Right-Angled):

    Since it's a right-angled triangle, we can use the Pythagorean Theorem: c2=a2+b2c^2 = a^2 + b^2 Given c=17c = 17 cm and a=15a = 15 cm, we substitute into the equation: 172=152+b217^2 = 15^2 + b^2 289=225+b2289 = 225 + b^2 b2=289225=64b^2 = 289 - 225 = 64 b=64=8 cmb = \sqrt{64} = 8 \text{ cm}

  2. For the Right Triangle (Non-Right-Angled):

    We use the Cosine Rule to find the unknown side aa: a2=b2+c22bccos(α)a^2 = b^2 + c^2 - 2bc \cdot \cos(\alpha) Given b=8b = 8 cm, c=10c = 10 cm, and α=60\alpha = 60^\circ, we substitute into the equation: a2=82+1022810cos(60)a^2 = 8^2 + 10^2 - 2 \cdot 8 \cdot 10 \cdot \cos(60^\circ) a2=64+1002810(12)a^2 = 64 + 100 - 2 \cdot 8 \cdot 10 \cdot \left(\frac{1}{2}\right) a2=64+10080a^2 = 64 + 100 - 80 a2=84a^2 = 84 a=849.17 cma = \sqrt{84} \approx 9.17 \text{ cm}

Summary of Answers:

  • For the left triangle, b=8b = 8 cm.
  • For the right triangle, a9.17a \approx 9.17 cm.

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Related Questions

  1. How would you find the area of each triangle?
  2. What is the perimeter of the right triangle after finding all sides?
  3. How does the Cosine Rule change if the angle α\alpha is obtuse?
  4. Can you calculate the angles of the left triangle using trigonometric ratios?
  5. What happens if the angle α\alpha in the right triangle was given as 9090^\circ?

Tip

Always remember to check whether a triangle is right-angled before choosing the appropriate theorem or formula for solving side lengths.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Geometry
Pythagorean Theorem
Cosine Rule

Formulas

Pythagorean Theorem: c^2 = a^2 + b^2
Cosine Rule: a^2 = b^2 + c^2 - 2bc * cos(α)

Theorems

Pythagorean Theorem
Cosine Rule

Suitable Grade Level

Grades 8-10