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Solve the Right Triangle: A Trigonometry Brain Teaser

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Solving Right Triangles: A Trigonometry Brain Teaser

Let’s dive into the exciting world of trigonometry with a brain-tickling problem involving a right triangle and an inscribed square. This challenge will test your understanding of trigonometric ratios and problem-solving skills.

The Challenge

Imagine a right triangle with sides of length 3, 4, and 5 units. Now, picture a square perfectly inscribed within this triangle, with all four vertices touching the sides of the triangle. Your mission is to determine the side length of this square.

Don’t worry if this seems daunting! We’ll break it down step by step.

Visualizing the Problem

To make things clearer, let’s draw a diagram:

Right triangle with inscribed square

Let’s label the vertices of the triangle as A, B, and C, with the right angle at C. The inscribed square will have vertices D, E, F, and G, where D lies on AC, E lies on BC, F lies on AB, and G lies on the hypotenuse AB.

Using Trigonometry to Solve

We can use trigonometric ratios to find the side length of the square. Let’s call the side length of the square ‘s’.

Consider triangle ABC. We know that:

  • AC = 3 units
  • BC = 4 units
  • AB = 5 units (hypotenuse)

Now, focus on triangle ADC. The angle DAC is the same as angle BAC, and we know the hypotenuse AD (which is equal to the side length of the square, ‘s’).

Using the cosine function, we have:

cos(angle BAC) = AC / AB = 3 / 5

In triangle ADC, we can write:

cos(angle DAC) = DC / AD = (3 – s) / s

Since angle BAC = angle DAC, we can equate the two expressions:

(3 – s) / s = 3 / 5

Solving for ‘s’, we get:

5(3 – s) = 3s

15 – 5s = 3s

15 = 8s

s = 15 / 8

Therefore, the side length of the square is 15/8 units.

Key Points to Remember

  • Understanding trigonometric ratios (sine, cosine, tangent) is crucial for solving problems involving right triangles.
  • Visualizing the problem with a clear diagram helps in identifying the relevant triangles and angles.
  • Break down complex problems into smaller, manageable steps.

Practice Makes Perfect

This was just one example of a trigonometry brain teaser. Try solving other problems involving inscribed shapes within right triangles. The more you practice, the better you’ll become at applying trigonometry in various scenarios.

Happy problem-solving!

Inscribed Shapes