Trigonometry Pile Up! How long is this side? 1.7 cm 1.7 cm 9 4.3 cm 21 71° 2.2 cm 53° 2.1 cm 3.6 cm 6 37° 42° 3.2 cm 11° 2 34° 8 cm 2.9 cm 2.5 cm 3.8 cm © 2012 www.greatmathsteachingid
The Correct Answer and Explanation is:
Based on the provided image, the correct length of the side in question is 6.39 cm.
Explanation
This trigonometry problem asks for the total length of the top horizontal side of the stacked triangles. This total length can be found by calculating the lengths of the horizontal sides of the two top-most triangles and adding them together. The other triangles and measurements further down the “pile up” are extraneous details not required for the solution.
First, let’s calculate the length of the horizontal side of the top-right triangle. This is a right-angled triangle with a given angle of 71 degrees. The vertical side, which is adjacent to the 71-degree angle, is given as 2.2 cm. The horizontal side we need to find is opposite this angle. We can use the tangent trigonometric function, which relates the opposite and adjacent sides:
tan(angle) = Opposite / Adjacent
tan(71°) = Right Horizontal Side / 2.2 cm
Right Horizontal Side = 2.2 * tan(71°)
Right Horizontal Side ≈ 2.2 * 2.9042 ≈ 6.389 cm
Next, we calculate the length of the horizontal side of the top-left triangle. This is also a right-angled triangle. Careful observation of the labels reveals a key detail. The hypotenuse of this triangle is labeled as 1.7 cm. The vertical leg of this same triangle is also labeled as 1.7 cm. We can use the Pythagorean theorem (a² + b² = c²) to find the length of the remaining horizontal side (a):
a² + (1.7)² = (1.7)²
a² = (1.7)² – (1.7)²
a² = 0
a = 0 cm
This means the horizontal side of the top-left triangle has a length of zero.
Finally, to find the total length requested, we add the lengths of the two horizontal segments:
Total Length = (Right Horizontal Side) + (Left Horizontal Side)
Total Length = 6.389 cm + 0 cm = 6.389 cm
Rounding to two decimal places, the final answer is 6.39 cm.
