What is the missing value in the triangle puzzle shown in the image?

Question image

Understand the Problem

The question is asking for assistance in solving a mathematical problem involving triangles with certain values given. The user seeks to find the missing value represented by a question mark in one of the triangles.

Answer

The missing angle is approximately $56.6^\circ$.
Answer for screen readers

The missing angle ( ? ) is approximately ( 56.6^\circ ).

Steps to Solve

  1. Identifying the relationship between the sides and angles of the triangles

In the given problem, the triangles appear to include values that relate to their angles (28° and 50°) and the lengths of their sides (2, 3; 4, 5; 3, 5). Observe the relationships to establish a pattern.

  1. Analyzing the first triangle

For the left triangle:

  • The two inner angles are represented as (28^\circ) and the corresponding sides are 2 and 3.

Using the sine rule:

$$ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} $$

This helps to establish the ratio of sides to angles.

  1. Setting up the second triangle

Looking at the second triangle (middle) with angles and sides of 50°, 4, and 5. Using the sine rule again:

  • Let the opposite angle to side 4 be (x):

$$ \frac{4}{\sin(50^\circ)} = \frac{5}{\sin(x)} $$

  1. Finding the unknown angle in the third triangle

In the third triangle, we need to determine the angle opposite to the side with (?).

Using the sine relationship from the previous triangles, we evaluate:

$$ \frac{3}{\sin(28^\circ)} = \frac{5}{\sin(?)} $$

  1. Calculating the unknown angle

Rearranging the equation:

$$ \sin(?) = \frac{5 \cdot \sin(28^\circ)}{3} $$

Now, calculate the value of ( \sin(?) ).

  1. Final Calculation Using Inverse Function

Using the inverse sine function, find the angle:

$$ ? = \sin^{-1}\left(\frac{5 \cdot \sin(28^\circ)}{3}\right) $$

The missing angle ( ? ) is approximately ( 56.6^\circ ).

More Information

The calculation shows how the sides and angles relate in triangles. By utilizing the sine rule, we can solve for unknown angles effectively, illustrating important properties of triangles.

Tips

  • Confusing angle measures with side lengths. It's crucial to keep different types of measurements distinct.
  • Not using the sine rule correctly, especially when converting between degrees and sides.

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