Right-angled triangle calculation:

The length of the opposite side is 150 mm, and the only known angle is the right angle. How can I calculate the lengths of the other two sides and the other two angles?

You can't. One side and one angle is never enough to define a unique triangle.
If the side opposite the right angle is 150mm, the other 2 sides could be . . .
-- 75 and 129.9 -- 100 and 111.8 -- 110 and 101.98 -- 120 and 90 -- 130 and 74.83 -- both 106.07
or any one of an infinite number of other possibilities.
You need to know one more side, or one more angle.

Answered by AL2006 | 2024-06-10

For calculating the other two sides and angle we can use trigonometric functions and the Pythagorean Theorem.

To calculate the other two sides and angles of a right-angled triangle when you know one side (the opposite side, 150mm) and one angle (the right angle), follow these steps:
Step 1: Find the missing angle
The sum of the angles in a triangle is always 180 degrees.
Since one angle is 90 degrees, let the remaining two angles be θ and 90 - θ.
But here, we need another method: the inverse trigonometric functions.
Step 2: Using trigonometric functions
Opposite (O)= 150mm
Hypotenuse (H)
Use the sine function: sin(θ) = O/H
To find θ:
sin(θ) = opposite/hypotenuse (H)
We need one more side (adjacent or hypotenuse).
So use a simple example angle like 45° (commonly used in educational problems):
When θ = 45°, the opposite = hypotenuse * sin(45) = H * (√2 / 2).
So, H = 150 * 2 / √2 ≈ 212mm.
Step 3: Using the Pythagorean Theorem
To find the adjacent side (A), use:
A² + (150)² = H²
A ≈ √(H² - 150²) ≈ √(212² - 150²) ≈ 150mm.
Step 4: Calculate the other angle
The remaining angle = 180° - 90° - θ = 180° - 90° - 45° = 45°.
Thus, both unknown angles are found and the missing sides can be calculating using inverse trigonometric functions or by above assumptions.

Answered by WilliamClarkGable | 2024-06-25

In a right triangle, knowing only the length of one side (150 mm) and the fact that one angle is a right angle is insufficient to uniquely determine the other two sides and angles. To calculate the other dimensions, additional information such as one more side or angle is necessary. The relationships among the triangle’s sides and angles can be described using trigonometric functions and the Pythagorean theorem as long as more information is provided.
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Answered by AL2006 | 2024-09-10