The Hypotenuse Of A Right Triangle Measures 18 Cm And One Of Its Legs Measures 10 Cm. Find The Measure
Understanding the properties of right triangles is fundamental in geometry, especially when solving for unknown sides using the Pythagorean theorem. In this article, we will explore how to determine the length of the missing leg in a right triangle when given the hypotenuse and one leg. The specific problem states: The hypotenuse of a right triangle measures 18 cm, and one of its legs measures 10 cm. Find the measure of the other leg.
This problem provides an excellent opportunity to review the Pythagorean theorem, apply algebraic techniques, and understand the geometric principles behind right triangles.
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Understanding the Right Triangle and Its Components
Before solving the problem, it is essential to understand the components involved:
- Right Triangle: A triangle with one 90-degree angle.
- Hypotenuse (c): The side opposite the right angle, and the longest side of the triangle.
- Legs (a and b): The two sides forming the right angle.
Given:
- Hypotenuse \( c = 18\, \text{cm} \)
- One leg \( a = 10\, \text{cm} \)
- The other leg \( b \) is unknown and needs to be found.
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Applying the Pythagorean Theorem
The Pythagorean theorem states that in a right triangle:
\[ a^2 + b^2 = c^2 \]
Where:
- \( a \) and \( b \) are the legs,
- \( c \) is the hypotenuse.
Given the values:
\[ 10^2 + b^2 = 18^2 \]
Calculating the squares:
\[ 100 + b^2 = 324 \]
Now, isolate \( b^2 \):
\[ b^2 = 324 - 100 \]
\[ b^2 = 224 \]
To find \( b \), take the square root of both sides:
\[ b = \sqrt{224} \]
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Calculating the Missing Leg
The square root of 224 can be simplified:
- Prime factorization of 224:
- Expressed as prime factors:
- Simplify the square root:
- Simplify further:
Therefore:
\[ b = 4 \times \sqrt{14} \, \text{cm} \]
For an approximate decimal value:
\[ \sqrt{14} \approx 3.7417 \]
\[ b \approx 4 \times 3.7417 \approx 14.9668\, \text{cm} \]
Final answer:
\[ \boxed{
b \approx 14.97\, \text{cm}
} \]
The measure of the missing leg is approximately 14.97 cm.
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Step-by-Step Solution Summary
To summarize the steps involved in solving for the missing leg:
- Identify the known values: hypotenuse \( c = 18\, \text{cm} \), one leg \( a = 10\, \text{cm} \).
- Apply the Pythagorean theorem: \( a^2 + b^2 = c^2 \).
- Substitute known values: \( 100 + b^2 = 324 \).
- Solve for \( b^2 \): \( b^2 = 224 \).
- Find \( b \) by calculating \( \sqrt{224} \), simplified as \( 4 \sqrt{14} \).
- Approximate the decimal value for practical use: approximately 14.97 cm.
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Additional Tips for Solving Right Triangle Problems
When approaching similar problems, consider the following:
- Always verify the given data: Ensure the hypotenuse is the longest side.
- Identify known and unknown components: Clearly distinguish between what is given and what is to be found.
- Use the Pythagorean theorem carefully: Substitute known values and perform algebraic operations step-by-step.
- Simplify square roots when possible: Look for perfect squares to simplify radical expressions.
- Use a calculator for decimal approximations: Especially when dealing with non-perfect squares.
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Real-Life Applications of Finding Missing Sides in Right Triangles
Understanding how to find missing sides in right triangles has numerous practical applications:
- Architecture and Engineering: Calculating lengths of structural components.
- Navigation and GPS: Determining distances when traveling diagonally.
- Computer Graphics: Calculating pixel distances and diagonals.
- Physics: Resolving vectors into components.
- Everyday Problem Solving: For example, determining the length of a ladder needed to reach a certain height when placed at a specific distance from the wall.
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Practice Problems to Reinforce Understanding
To solidify your grasp of right triangle problems, try solving these:
- In a right triangle, the hypotenuse measures 25 cm, and one leg measures 15 cm. Find the other leg.
- A right triangle has legs of 8 cm and 15 cm. What is the length of the hypotenuse?
- If the hypotenuse of a right triangle is 20 meters, and one leg is 12 meters, what is the length of the other leg?
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Conclusion
Determining the missing side in a right triangle when given the hypotenuse and one leg is a fundamental skill in geometry, rooted in the Pythagorean theorem. In the specific problem where the hypotenuse measures 18 cm and one leg measures 10 cm, the other leg measures approximately 14.97 cm. Understanding these concepts not only enhances your mathematical problem-solving skills but also prepares you for real-world applications where spatial reasoning is essential.
By following systematic steps—applying the Pythagorean theorem, simplifying radicals, and performing accurate calculations—you can confidently solve similar problems with ease. Keep practicing with different values and scenarios to strengthen your understanding of right triangle properties and the Pythagorean theorem.
