# How to Solve Two Triangles with Pythagoras’ Theorem

Pythagoras’ Theorem is a fundamental concept in mathematics that has numerous applications in geometry, trigonometry, and real-world problem-solving. One of the most interesting and challenging applications of this theorem is solving problems involving two triangles. In this article, we will explore how to use Pythagoras’ Theorem to solve two triangles and provide step-by-step guidance to help you master this essential skill.

## Understanding Pythagoras’ Theorem

Before diving into solving two triangles, let’s briefly review Pythagoras’ Theorem and its basic concepts.

### The Theorem Statement

Pythagoras’ Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (known as the base and perpendicular).

Mathematically, this can be expressed as:

\(a^2 + b^2 = c^2\)

Where:

- \(a\) and \(b\) are the lengths of the base and perpendicular
- \(c\) is the length of the hypotenuse

### Applying the Theorem

To apply Pythagoras’ Theorem, you need to identify the right-angled triangle within a given problem and determine which sides correspond to the hypotenuse, base, and perpendicular. Once you have this information, you can use the theorem to calculate the missing side lengths or solve for other unknowns in the problem.

## Solving Two Triangles with Pythagoras’ Theorem

Now that we have a solid understanding of Pythagoras’ Theorem, let’s explore how to use it to solve problems involving two triangles.

### Identifying the Triangles and Shared Side

The first step in solving two triangles with Pythagoras’ Theorem is to identify the two right-angled triangles within the problem. Often, these triangles will share a common side, which can be used to link the triangles and solve for missing lengths or angles.

Look for the shared side and label it accordingly. This shared side will play a crucial role in solving the problem, as it will allow you to establish relationships between the two triangles.

### Applying Pythagoras’ Theorem to Each Triangle

Once you have identified the two right-angled triangles and their shared side, apply Pythagoras’ Theorem to each triangle separately. This will help you determine the missing side lengths or establish relationships between the sides of the triangles.

For example, let’s consider a problem where two right-angled triangles share a common hypotenuse. We can label the sides of the first triangle as \(a_1\), \(b_1\), and \(c\), and the sides of the second triangle as \(a_2\), \(b_2\), and \(c\).

Applying Pythagoras’ Theorem to each triangle, we get:

First triangle: \(a_1^2 + b_1^2 = c^2\)

Second triangle: \(a_2^2 + b_2^2 = c^2\)

### Establishing Relationships and Solving

With the equations from applying Pythagoras’ Theorem to each triangle, you can now establish relationships between the sides and solve for the missing lengths or angles.

In the example above, since both triangles share the same hypotenuse \(c\), we can equate the right-hand sides of the equations:

\(a_1^2 + b_1^2 = a_2^2 + b_2^2\)

This relationship allows us to solve for missing side lengths or determine the relationships between the sides of the two triangles.

### Solving for Missing Angles

In some problems, you may need to solve for missing angles in addition to side lengths. Once you have determined the side lengths using Pythagoras’ Theorem, you can use trigonometric ratios (sine, cosine, or tangent) to calculate the missing angles.

For example, if you know the lengths of the hypotenuse \(c\) and the base \(a\) in a right-angled triangle, you can calculate the angle \(\theta\) opposite the base using the inverse cosine function:

\( \displaystyle \theta = \cos^{-1}\frac{a}{c}\)

By applying trigonometric ratios to the sides you calculated using Pythagoras’ Theorem, you can solve for missing angles in the two triangles.

## Example Problem

Let’s apply the steps we’ve discussed to solve a problem involving two triangles using Pythagoras’ Theorem.

Problem: Two right-angled triangles share a common hypotenuse of length 25 units. The base of the first triangle is 7 units longer than the base of the second triangle. The perpendicular of the second triangle is 4 units longer than the perpendicular of the first triangle. Find the lengths of the bases and perpendiculars of both triangles.

Solution:

Step 1: Identify the triangles and shared sides.

- The two right-angled triangles share a common hypotenuse of length 25 units.

Step 2: Label the sides of the triangles.

- First triangle: base \(a_1\), perpendicular \(b_1\), hypotenuse \(c\)
- Second triangle: base \(a_2\), perpendicular \(b_2\), hypotenuse \(c\)

Step 3: Express the relationships between the sides.

- \(a_1 = a_2 + 7\)
- \(b_2 = b_1 + 4\)
- \(c = 25\)

Step 4: Apply Pythagoras’ Theorem to each triangle.

- First triangle: \(a_1^2 + b_1^2 = c^2\)
- Second triangle: \(a_2^2 + b_2^2 = c^2\)

Step 5: Substitute the relationships and solve.

- \((a_2 + 7)^2 + b_1^2 = 25^2\)
- \(a_2^2 + (b_1 + 4)^2 = 25^2\)

Expanding and simplifying the equations, we get:

- \(a_2^2 + 14a_2 + 49 + b_1^2 = 625\)
- \(a_2^2 + b_1^2 + 8b_1 + 16 = 625\)

Subtracting the second equation from the first:

- \(14a_2 + 49 – 8b_1-16 = 0\)
- \(14a_2-8b_1 + 33 = 0\)

Solving for \(a_2\) in terms of \(b_1\):

- \( \displaystyle a_2 = \frac{8b_1-33}{14}\)

Substituting this expression into the first equation and solving for \(b_1\):

- \( \displaystyle \left(\frac{8b_1-33}{14}\right)^2 + b_1^2 = 576\)
- \(b_1 = 12\) or \(b_1 = -9\) (rejected as length cannot be negative)

Substituting \(b_1 = 12\) into the expression for \(a_2\):

- \( \displaystyle a_2 = \frac{8 \times 12-33}{14} = \frac{63}{14} = 4.5\)

Therefore, the lengths of the bases and perpendiculars are:

- First triangle: base \(a_1 = 11.5\) units, perpendicular \(b_1 = 12\) units
- Second triangle: base \(a_2 = 4.5\) units, perpendicular \(b_2 = 16\) units

## Conclusion

Solving two triangles using Pythagoras’ Theorem may seem challenging at first, but by following the step-by-step approach outlined in this article, you can master this essential skill. Remember to identify the right-angled triangles, label their sides, and establish relationships between them. By applying Pythagoras’ Theorem to each triangle and solving the resulting equations, you can determine the missing side lengths and angles.

Practice is key to becoming proficient in solving two triangles with Pythagoras’ Theorem. Seek out various problems and work through them methodically, applying the techniques you’ve learned. With dedication and perseverance, you’ll soon find yourself confidently tackling even the most challenging two-triangle problems.

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