Math Problem Statement

How do you find the hypotenuse and angle in a right triangle with side lengths of 10 meters and 6 meters?

Solution

This image shows a right triangle with two side lengths labeled: one side is 10 meters (likely the height), and the base is 6 meters. There's also an angle marked at the base, likely intended to find either the angle or the hypotenuse length.

To solve for any unknown values, we could:

  1. Find the length of the hypotenuse using the Pythagorean theorem: c=(102+62)c = \sqrt{(10^2 + 6^2)}

  2. Calculate the angle opposite the height (10 meters) using trigonometry (sine, cosine, or tangent):

    • For the angle θ\theta opposite the 10-meter side, use: tan(θ)=106\tan(\theta) = \frac{10}{6}
    • Then, θ=tan1(106)\theta = \tan^{-1} \left( \frac{10}{6} \right).

Would you like me to proceed with these calculations or explore other aspects of this triangle?

Here are five related questions that expand upon this information:

  1. How do you calculate the area of a right triangle given two sides?
  2. What trigonometric function would be useful if only the hypotenuse and one side were known?
  3. How can the sine and cosine functions help find the missing sides if an angle is known?
  4. What happens to the angle measures if the triangle’s proportions are changed while keeping it similar?
  5. How would you find the length of the hypotenuse if only one leg and the angle were given?

Tip: Always double-check which side lengths correspond to the hypotenuse, opposite, and adjacent sides when using trigonometric ratios in a right triangle.

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Math Problem Analysis

Mathematical Concepts

Geometry
Right Triangles
Trigonometry

Formulas

Pythagorean theorem: c = √(a² + b²)
Tangent ratio: tan(θ) = opposite / adjacent

Theorems

Pythagorean theorem
Trigonometric ratios

Suitable Grade Level

Grades 8-10