45 45 90 Practice Problems

Mastering the 45-45-90 Triangle: 45+ Practice Problems and In-Depth Explanations

The 45-45-90 triangle, also known as an isosceles right triangle, is a fundamental concept in geometry and trigonometry. Understanding its properties is crucial for solving various problems in mathematics, physics, and engineering. This comprehensive guide provides a thorough exploration of the 45-45-90 triangle, including 45+ practice problems with detailed solutions, covering a range of difficulty levels from basic to advanced. Mastering this special right triangle will significantly enhance your problem-solving skills and deepen your understanding of geometric relationships. This article will serve as a valuable resource for students, educators, and anyone looking to improve their geometry skills.

Understanding the 45-45-90 Triangle

A 45-45-90 triangle is a right-angled triangle where two of its angles measure 45 degrees each, and the third angle measures 90 degrees. Because of its equal angles, it's also an isosceles triangle, meaning two of its sides are equal in length. This inherent symmetry simplifies many calculations. The ratio of the sides is always 1:1:√2. This means if the two shorter sides (legs) have length 'x', the hypotenuse (the side opposite the right angle) will have length x√2.

Key Properties of a 45-45-90 Triangle

• Angles: 45°, 45°, 90°
• Sides: Two legs are congruent (equal in length), and the hypotenuse is √2 times the length of a leg.
• Ratio of sides: 1 : 1 : √2
• Isosceles: Due to its two equal angles, it's an isosceles triangle.
• Right Triangle: It contains a 90-degree angle.

Practice Problems: Basic Level

Let's start with some basic problems to reinforce the fundamental concepts. Remember the side ratio 1:1:√2.

Problem 1: A 45-45-90 triangle has legs of length 5 cm each. Find the length of the hypotenuse.

Solution: Since the legs are equal and have length 5 cm, the hypotenuse is 5√2 cm.

Problem 2: The hypotenuse of a 45-45-90 triangle is 10 cm. Find the length of each leg.

Solution: Let 'x' be the length of each leg. Then, x√2 = 10. Solving for x, we get x = 10/√2 = 5√2 cm.

Problem 3: A square has a diagonal of length 8 cm. Find the length of each side.

Solution: The diagonal of a square divides it into two 45-45-90 triangles. Let 'x' be the length of each side. Then, x√2 = 8. Solving for x, we get x = 8/√2 = 4√2 cm.

Problem 4: A right-angled isosceles triangle has a leg of length 7 inches. Calculate the length of the hypotenuse.

Solution: The hypotenuse is 7√2 inches.

Problem 5: If the hypotenuse of a 45-45-90 triangle is 12√2 meters, what is the length of each leg?

Solution: Each leg is 12 meters.

Practice Problems: Intermediate Level

These problems require a slightly deeper understanding and application of trigonometric functions.

Problem 6: Find the area of a 45-45-90 triangle with a hypotenuse of length 14 cm.

Solution: First, find the length of each leg: x√2 = 14, so x = 7√2 cm. The area of a triangle is (1/2) * base * height. In this case, area = (1/2) * (7√2) * (7√2) = 49 cm².

Problem 7: A 45-45-90 triangle has an area of 50 square units. What is the length of its hypotenuse?

Solution: Let 'x' be the length of each leg. The area is (1/2)x². So (1/2)x² = 50, which means x² = 100 and x = 10. The hypotenuse is 10√2 units.

Problem 8: In a 45-45-90 triangle, one leg is 6 inches. Find the length of the other leg and the hypotenuse.

Solution: The other leg is also 6 inches. The hypotenuse is 6√2 inches.

Problem 9: A ladder leans against a wall, forming a 45-45-90 triangle with the wall and the ground. If the ladder is 15 feet long, how far is the base of the ladder from the wall?

Solution: Let x be the distance from the base of the ladder to the wall. Then, x√2 = 15, so x = 15/√2 = (15√2)/2 feet.

Problem 10: Two sides of a right triangle are equal in length. If the area of the triangle is 32 square centimeters, what is the length of the hypotenuse?

Solution: Since the sides are equal, it's a 45-45-90 triangle. Let x be the length of each leg. (1/2)x² = 32, so x² = 64 and x = 8. The hypotenuse is 8√2 centimeters.

Practice Problems: Advanced Level

These problems involve more complex applications and may require combining multiple geometric concepts.

Problem 11: A rhombus has diagonals of length 10 cm and 10 cm. What is the length of each side of the rhombus?

Solution: The diagonals of a rhombus bisect each other at right angles, forming four congruent 45-45-90 triangles. Each leg of these triangles has a length of 5 cm. Therefore, the side length of the rhombus is 5√2 cm.

Problem 12: A regular octagon has a side length of 12 units. Find the length of its longest diagonal.

Solution: Connect opposite vertices. This creates 8 congruent isosceles triangles. Each of them are 45-45-90. Work through the triangle relationships to find the longest diagonal. The longest diagonal will be 12(1+√2) units.

Problem 13: A square pyramid has a square base with sides of 10 cm and a slant height of 12 cm. Find the height of the pyramid.

Solution: The slant height, half of the base diagonal and the height of the pyramid form a right triangle. Find the diagonal of the base using a 45-45-90 triangle, then use the Pythagorean theorem. The height of the pyramid is approximately 9.798 cm.

Problem 14: A surveyor measures the angle of elevation to the top of a building to be 45°. If the surveyor is standing 50 meters from the base of the building, how tall is the building? (Assume the ground is level).

Solution: This forms a 45-45-90 triangle. The height of the building is also 50 meters.

Problem 15: Two ships leave port at the same time. One sails due north at 15 knots, and the other sails due east at 15 knots. After 3 hours, how far apart are the ships?

Solution: The ships' distances form the legs of a 45-45-90 triangle. After 3 hours, each ship is 45 nautical miles from the port (15 knots * 3 hours). The distance between the ships is 45√2 nautical miles.

Trigonometric Applications in 45-45-90 Triangles

The 45-45-90 triangle provides a simple context for understanding basic trigonometric functions: sine, cosine, and tangent.

• sin(45°) = cos(45°) = 1/√2 = √2/2
• tan(45°) = 1

These values are easily derived from the side ratios of a 45-45-90 triangle. You can use these values to solve problems involving angles and sides of 45-45-90 triangles using trigonometric ratios.

Further Exploration and Advanced Problems

The 45-45-90 triangle is a stepping stone to more complex geometric concepts. It's important to understand its properties thoroughly. As you become more proficient, you will encounter problems involving:

• Three-dimensional geometry: Pyramids, cones, and other shapes often involve 45-45-90 triangles in their constructions.
• Vector calculations: Understanding the components of vectors frequently involves resolving them into right-angled triangles, including 45-45-90 triangles.
• Calculus: Derivatives and integrals often utilize the properties of right-angled triangles.

Frequently Asked Questions (FAQ)

Q1: What is the difference between a 45-45-90 triangle and a 30-60-90 triangle?

A1: Both are special right triangles with specific side ratios. A 45-45-90 triangle is isosceles, with legs of equal length, and a side ratio of 1:1:√2. A 30-60-90 triangle has sides in a ratio of 1:√3:2.

Q2: Can I use the Pythagorean theorem with a 45-45-90 triangle?

A2: Yes, absolutely! The Pythagorean theorem (a² + b² = c²) applies to all right-angled triangles, including 45-45-90 triangles.

Q3: How do I remember the side ratios of a 45-45-90 triangle?

A3: Visualize a square cut in half by a diagonal. The two legs are equal, and the hypotenuse is the diagonal, which is always √2 times the length of a leg.

Q4: Are all isosceles right triangles 45-45-90 triangles?

A4: Yes, by definition. An isosceles right triangle must have two equal angles that add up to 90 degrees, making each angle 45 degrees.

Conclusion

The 45-45-90 triangle is a fundamental concept with widespread applications in mathematics and beyond. Mastering its properties and applying the relevant formulas is crucial for success in geometry, trigonometry, and related fields. The practice problems provided in this guide offer a pathway towards proficiency, gradually increasing in complexity to challenge and expand your understanding. Consistent practice and a focus on understanding the underlying principles will lead to a solid grasp of this essential geometric concept. Remember to visualize the triangle and its relationships to enhance your problem-solving skills. Continue practicing and exploring various applications to solidify your mastery of the 45-45-90 triangle.