What is the length of the side opposite angle b? 3 units 4 units 5 units 6 units

Explanation

Identifying the right triangle

• The question "what is the length of the side opposite angle b?" implies a right triangle with sides 3, 4, and 5 units
• This is a well-known 3-4-5 right triangle, which satisfies the Pythagorean theorem

Applying the Pythagorean theorem

• The Pythagorean theorem states: a² + b² = c², where c is the hypotenuse
• In this case: 3² + 4² = 5² (9 + 16 = 25), confirming the 3-4-5 triangle side lengths
• The hypotenuse is always the longest side of a right triangle

Determining the side opposite angle b

• The side opposite the right angle (90°) is always the hypotenuse
• In a right triangle, the largest angle is opposite the longest side
• Therefore, angle b must be opposite the 5-unit side, as it's the longest among the options of 3 units, 4 units, 5 units, and 6 units

Verification using trigonometric ratios

• We can verify the side length opposite angle b in a right triangle using the sine ratio: sin(angle) = opposite / hypotenuse
• For the 3-4-5 triangle: sin(b) = 5/5 = 1, which corresponds to a 90° angle
• This confirms that the 5-unit side is indeed opposite to angle b (the right angle) in the 3-4-5 triangle side lengths

FAQ

What is the length of the side opposite angle b in a 3-4-5 right triangle?

The length of the side opposite angle b in a 3-4-5 right triangle is 5 units.

How do you determine which side is opposite angle b in a right triangle?

In a right triangle, the side opposite angle b is the hypotenuse, which is always the longest side. In a 3-4-5 triangle, this is the 5-unit side.

Why is 5 units the correct answer for the side length opposite angle b?

5 units is the correct answer because it's the longest side in the 3-4-5 triangle, and the longest side is always opposite the largest angle (90°) in a right triangle.

What are the side lengths of a 3-4-5 triangle?

The side lengths of a 3-4-5 triangle are 3 units, 4 units, and 5 units, with 5 units being the length of the hypotenuse.

How can you verify that 5 units is the correct side length opposite angle b?

You can verify using the Pythagorean theorem (a² + b² = c²) or by using trigonometric ratios, such as sine, where sin(b) = opposite / hypotenuse = 5/5 = 1, corresponding to a 90° angle.