Math Problem Statement

Use the right triangle and the given information to solve the triangle.

a equals 4 commaa=4,

c equals 10c=10​;

find b comma Upper A comma and Upper B

Solution

To solve the right triangle with the given information:

Given:

  • a = 4 (leg)
  • c = 10 (hypotenuse)

We need to find:

  1. The other leg b
  2. The angle A
  3. The angle B

Step 1: Find b using the Pythagorean Theorem:

c2=a2+b2c^2 = a^2 + b^2 102=42+b210^2 = 4^2 + b^2 100=16+b2100 = 16 + b^2 b2=10016=84b^2 = 100 - 16 = 84 b=849.17b = \sqrt{84} \approx 9.17

Step 2: Find angle A using trigonometric ratios:

sin(A)=ac\sin(A) = \frac{a}{c} sin(A)=410=0.4\sin(A) = \frac{4}{10} = 0.4 A=arcsin(0.4)23.58A = \arcsin(0.4) \approx 23.58^\circ

Step 3: Find angle B:

Since the sum of angles in a triangle is 9090^\circ for a right triangle: B=90AB = 90^\circ - A B9023.58=66.42B \approx 90^\circ - 23.58^\circ = 66.42^\circ

Final Results:

  1. b ≈ 9.17
  2. A ≈ 23.58°
  3. B ≈ 66.42°

Would you like a breakdown of how I used trigonometric functions or further steps? Let me know!


Related Questions:

  1. How do we derive the Pythagorean theorem?
  2. What are other methods to find angles in a right triangle?
  3. How does sine differ from cosine when solving triangles?
  4. What is the significance of complementary angles in right triangles?
  5. How would the calculation change if the hypotenuse was unknown?

Tip:

When working with right triangles, always verify your calculations by checking that the sum of the angles is 9090^\circ and the sides satisfy the Pythagorean theorem.

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Math Problem Analysis

Mathematical Concepts

Geometry
Trigonometry
Right Triangles

Formulas

Pythagorean Theorem: c^2 = a^2 + b^2
Sine function: sin(A) = opposite/hypotenuse
Complementary angles: A + B = 90°

Theorems

Pythagorean Theorem
Trigonometric Ratios

Suitable Grade Level

Grades 8-10