A door frame that appears rectangular has height 80 in., width 32 in., and one diagonal that measures 86in. Is the door frame actually rectangular?
only if the angles are right angles so that the diagonal satisfies the Pythagorean Theorem. That is, if
80^2 + 32^2 = 86^2
does it?
no
To determine if the door frame is actually rectangular, we can use the Pythagorean theorem to check if the given measurements are consistent. According to the theorem, in a right-angled triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
Let's calculate the length of the missing side using the given measurements:
height: 80 in.
width: 32 in.
diagonal: 86 in.
Using the Pythagorean theorem, we have:
(32^2 + 80^2) = diagonal^2
(1024 + 6400) = diagonal^2
7424 = diagonal^2
diagonal = √7424
diagonal ≈ 86.088
The calculated length of the diagonal is approximately 86.088 inches, which is close to the given length of 86 inches. Hence, the measurements are consistent, and the door frame is actually rectangular.
To determine if the door frame is actually rectangular, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the height (h) and the width (w) of the door frame form the two sides of a right-angled triangle, and the diagonal (d) forms the hypotenuse. We are given that the height is 80 in., the width is 32 in., and the diagonal measures 86 in.
We can set up the equation using the Pythagorean theorem:
d^2 = h^2 + w^2
Substituting the known values:
86^2 = 80^2 + 32^2
7396 = 6400 + 1024
7396 = 7424
Since the two sides of the equation do not match, we can conclude that the door frame is not actually rectangular.
Note: In this case, there is a discrepancy between the measurements provided and the Pythagorean relationship. It is possible that the door frame is not exactly rectangular, or there may be measurement errors. Double-checking the measurements or inspecting the door frame visually could help confirm the shape.