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Two students describe the sides of right triangle ABC in relation to $\angle B$.




Tomas		Iliana

$\overline{ AB }$ is the hypotenuse.		$\overline{ AB }$ is the hypotenuse.

$\overline{ AC }$ is the opposite side.		$\overline{ BC }$ is the opposite side.

$\overline{ BC }$ is the adjacent side.		$\overline{ AC }$ is the adjacent side.






Who is correct? Explain.

Verified Answer :

  • Identify the hypotenuse as the side opposite the right angle (\(\overline{AB}\)).
  • Recognize that the side opposite \(\angle B\) is \(\overline{AC}\) and the side adjacent to \(\angle B\) is \(\overline{BC}\).
  • Compare these definitions with Tomas's and Iliana's descriptions to confirm that Tomas is correct.

Explanation

  1. Analysis of the sides of right triangle ABC relative to angle B

In a right triangle \(ABC\), the sides are defined by their relationship to the vertices and angles. The side opposite to a vertex is traditionally denoted by the corresponding lowercase letter: \(a\) is opposite \(A\) (side \(\overline{BC}\)), \(b\) is opposite \(B\) (side \(\overline{AC}\)), and \(c\) is opposite \(C\) (side \(\overline{AB}\)).

Given that \(\overline{AB}\) is the hypotenuse, the right angle must be at vertex \(C\). Therefore, we analyze the sides in relation to \(\angle B\):

  • The hypotenuse is the side opposite the right angle, which is \(\overline{AB}\).
  • The opposite side to \(\angle B\) is the side that does not touch vertex \(B\), which is \(\overline{AC}\).
  • The adjacent side to \(\angle B\) is the side that forms part of \(\angle B\) but is not the hypotenuse, which is \(\overline{BC}\).

Comparing these definitions with the claims in the table:

  • Tomas states \(\overline{AB}\) is the hypotenuse, \(\overline{AC}\) is the opposite side, and \(\overline{BC}\) is the adjacent side. This matches our derivation.
  • Iliana states \(\overline{AB}\) is the hypotenuse, but incorrectly identifies \(\overline{BC}\) as the opposite side and \(\overline{AC}\) as the adjacent side.

Therefore, Tomas correctly identified the sides of the triangle.

Examples

Understanding the orientation of sides in a right triangle is crucial in trigonometry. For instance, if you are calculating the height of a building (the opposite side) using the angle of elevation (angle B) and the distance from the building (the adjacent side), you rely on the tangent function defined as \(\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AC}{BC}\). Correctly identifying these components ensures that your calculations for distances and heights are accurate and reliable in real-world surveying or construction projects.

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