Analyzing the Given Sets of Sides

Oct 13, 2025 by miodeya.com 130 views

1. From the given sets of sides, select by ticking the box, only those that can form a right-angled triangle (assume the sides are of the same unit of length).

Understanding the Basics of Right-Angled Triangles

A right-angled triangle is a triangle with one angle that is 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. In a right-angled triangle, the Pythagorean theorem states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): a^2 + b^2 = c^2.

Analyzing the Given Sets of Sides

We are given three sets of sides, and we need to determine which ones can form a right-angled triangle.

Set a: ${ $2, \sqrt{5}, \sqrt{3}\$}$

To determine if this set can form a right-angled triangle, we need to check if the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. Let's assume that the longest side is ${\sqrt{5}}$ , then we have:

${\sqrt{5}}$ ^2 = 2^2 + ${\sqrt{3}}$ ^2

5 = 4 + 3

5 = 7

This is not true, so ${\sqrt{5}}$ cannot be the hypotenuse. Now, let's assume that the longest side is ${\sqrt{3}}$ , then we have:

${\sqrt{3}}$ ^2 = 2^2 + ${\sqrt{5}}$ ^2

3 = 4 + 5

3 = 9

This is not true, so ${\sqrt{3}}$ cannot be the hypotenuse. Now, let's assume that the longest side is 2, then we have:

2^2 = ${\sqrt{5}}$ ^2 + ${\sqrt{3}}$ ^2

4 = 5 + 3

4 = 8

This is not true, so 2 cannot be the hypotenuse. Therefore, set a cannot form a right-angled triangle.

Set b: ${$ \sqrt{13}, 2, 3$}$

${\sqrt{13}}$ ^2 = 2^2 + 3^2

13 = 4 + 9

13 = 13

This is true, so ${\sqrt{13}}$ can be the hypotenuse. Therefore, set b can form a right-angled triangle.

Set c: ${$ \sqrt{3}, 2, \sqrt{7}$}$

${\sqrt{7}}$ ^2 = ${\sqrt{3}}$ ^2 + 2^2

7 = 3 + 4

7 = 7

This is true, so ${\sqrt{7}}$ can be the hypotenuse. Therefore, set c can form a right-angled triangle.

Conclusion

In conclusion, only sets b and c can form a right-angled triangle. Set a cannot form a right-angled triangle because the square of the longest side is not equal to the sum of the squares of the other two sides.

Real-World Applications of Right-Angled Triangles

Right-angled triangles have many real-world applications, including:

Building Design: Right-angled triangles are used in building design to ensure that the structure is stable and can withstand various loads.
Physics: Right-angled triangles are used in physics to calculate distances, velocities, and accelerations.
Engineering: Right-angled triangles are used in engineering to design and build bridges, roads, and other infrastructure.
Navigation: Right-angled triangles are used in navigation to calculate distances and directions.

Tips and Tricks for Working with Right-Angled Triangles

Here are some tips and tricks for working with right-angled triangles:

Use the Pythagorean Theorem: The Pythagorean theorem is a powerful tool for calculating the length of the hypotenuse of a right-angled triangle.
Use Trigonometry: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It can be used to calculate the length of the sides of a right-angled triangle.
Use Geometry: Geometry is the branch of mathematics that deals with the properties and relationships of points, lines, and shapes. It can be used to calculate the length of the sides of a right-angled triangle.

Common Mistakes to Avoid When Working with Right-Angled Triangles

Here are some common mistakes to avoid when working with right-angled triangles:

Not Using the Pythagorean Theorem: The Pythagorean theorem is a powerful tool for calculating the length of the hypotenuse of a right-angled triangle. Not using it can lead to incorrect results.
Not Checking the Hypotenuse: The hypotenuse is the longest side of a right-angled triangle. Not checking it can lead to incorrect results.
Not Using Trigonometry: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. Not using it can lead to incorrect results.

Conclusion

In conclusion, right-angled triangles are an important concept in mathematics and have many real-world applications. By understanding the basics of right-angled triangles and using the Pythagorean theorem, trigonometry, and geometry, you can calculate the length of the sides of a right-angled triangle and avoid common mistakes.
2. Frequently Asked Questions (FAQs) About Right-Angled Triangles

Q: What is a right-angled triangle?

A: A right-angled triangle is a triangle with one angle that is 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula that states that the square of the length of the hypotenuse (c) of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides (a and b): a^2 + b^2 = c^2.

Q: How do I use the Pythagorean theorem to calculate the length of the hypotenuse?

A: To use the Pythagorean theorem to calculate the length of the hypotenuse, you need to know the lengths of the other two sides. Then, you can plug the values into the formula: a^2 + b^2 = c^2. Solve for c to find the length of the hypotenuse.

Q: Can I use the Pythagorean theorem to calculate the length of the legs?

A: Yes, you can use the Pythagorean theorem to calculate the length of the legs. If you know the length of the hypotenuse and one of the legs, you can plug the values into the formula: a^2 + b^2 = c^2. Solve for a or b to find the length of the leg.

Q: What is the difference between a right-angled triangle and an isosceles triangle?

A: A right-angled triangle is a triangle with one angle that is 90 degrees. An isosceles triangle is a triangle with two sides of equal length. While a right-angled triangle can be isosceles, not all isosceles triangles are right-angled.

Q: Can I use trigonometry to calculate the length of the sides of a right-angled triangle?

A: Yes, you can use trigonometry to calculate the length of the sides of a right-angled triangle. Trigonometry deals with the relationships between the sides and angles of triangles. You can use trigonometric functions such as sine, cosine, and tangent to calculate the length of the sides.

Q: What is the relationship between the sine, cosine, and tangent of a right-angled triangle?

A: The sine, cosine, and tangent of a right-angled triangle are related to the lengths of the sides. The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: Can I use geometry to calculate the length of the sides of a right-angled triangle?

A: Yes, you can use geometry to calculate the length of the sides of a right-angled triangle. Geometry deals with the properties and relationships of points, lines, and shapes. You can use geometric properties such as congruence and similarity to calculate the length of the sides.

Q: What is the difference between a right-angled triangle and an obtuse triangle?

A: A right-angled triangle is a triangle with one angle that is 90 degrees. An obtuse triangle is a triangle with one angle that is greater than 90 degrees. While a right-angled triangle can be obtuse, not all obtuse triangles are right-angled.

Q: Can I use the Pythagorean theorem to calculate the length of the sides of an obtuse triangle?

A: No, you cannot use the Pythagorean theorem to calculate the length of the sides of an obtuse triangle. The Pythagorean theorem only applies to right-angled triangles.

Q: What is the relationship between the Pythagorean theorem and the law of cosines?

A: The Pythagorean theorem and the law of cosines are related but distinct concepts. The Pythagorean theorem applies to right-angled triangles, while the law of cosines applies to all triangles. The law of cosines states that the square of the length of a side of a triangle is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of the lengths of the other two sides and the cosine of the angle between them.

Q: Can I use the law of cosines to calculate the length of the sides of a right-angled triangle?

A: Yes, you can use the law of cosines to calculate the length of the sides of a right-angled triangle. The law of cosines is a more general formula that applies to all triangles, including right-angled triangles.

Q: What is the difference between a right-angled triangle and a scalene triangle?

A: A right-angled triangle is a triangle with one angle that is 90 degrees. A scalene triangle is a triangle with all sides of different lengths. While a right-angled triangle can be scalene, not all scalene triangles are right-angled.

Q: Can I use the Pythagorean theorem to calculate the length of the sides of a scalene triangle?

A: No, you cannot use the Pythagorean theorem to calculate the length of the sides of a scalene triangle. The Pythagorean theorem only applies to right-angled triangles.