Question

applying the converse of the pythagorean theorem
janice examines the given triangle and estimates the longest side has a length of 25 units—if it is a right triangle. how does her estimate compare to the actual length?
○ it is exactly correct.
○ it is under by approximately 0.6 units.
○ it is over by approximately 0.6 units.
○ it is over by 13 units.
(image of a triangle with sides 16 and 20, and the base side unlabeled)

Explanation:

Step1: Identify the triangle type

The triangle has sides 16 and 20, and we assume it's a right triangle (since we're applying the Pythagorean theorem). For a right triangle, the hypotenuse \( c \) is found using \( c = \sqrt{a^2 + b^2} \), where \( a = 16 \) and \( b = 20 \).

Step2: Calculate the actual hypotenuse

First, compute \( a^2 + b^2 \):
\( 16^2 + 20^2 = 256 + 400 = 656 \).
Then, take the square root: \( \sqrt{656} \approx 25.612 \)? Wait, no—wait, 16 and 20: wait, maybe I mixed up legs and hypotenuse? Wait, no, the longest side is the hypotenuse. Wait, Janice estimated 25. Let's recalculate: \( \sqrt{16^2 + 20^2} = \sqrt{256 + 400} = \sqrt{656} \approx 25.61 \)? Wait, no, that can't be. Wait, 16-20-? Wait, maybe the triangle is right-angled, so the two legs are 16 and 20? Wait, no, 16 and 20, then hypotenuse is \( \sqrt{16^2 + 20^2} = \sqrt{256 + 400} = \sqrt{656} \approx 25.61 \)? Wait, but Janice estimated 25. Wait, no, maybe I made a mistake. Wait, 16 and 20: let's check \( 16^2 + 20^2 = 256 + 400 = 656 \). \( \sqrt{656} \approx 25.61 \)? Wait, no, 25 squared is 625, 26 squared is 676. So \( \sqrt{656} \) is between 25 and 26. Let's compute it more accurately: \( 25.6^2 = 655.36 \), which is very close to 656. So \( \sqrt{656} \approx 25.6 \). Wait, Janice estimated 25. So the actual length is approximately 25.6, and her estimate is 25. So the difference is \( 25.6 - 25 = 0.6 \). Wait, no—wait, if Janice estimated 25, but the actual is ~25.6, then her estimate is under by ~0.6? Wait, no, wait: 25.6 - 25 = 0.6. So her estimate (25) is less than the actual (25.6), so it's under by approximately 0.6 units? Wait, no, wait, maybe I had the legs wrong. Wait, maybe the triangle is such that 16 and 20 are legs, and the hypotenuse is the longest side. Wait, but let's re-express:

Wait, the problem says "Janice examines the given triangle and estimates the longest side has a length of 25 units—if it is a right triangle. How does her estimate compare to the actual length?"

So, for a right triangle with legs 16 and 20, hypotenuse \( c = \sqrt{16^2 + 20^2} = \sqrt{256 + 400} = \sqrt{656} \approx 25.61 \). Wait, but 25.61 - 25 = 0.61, so her estimate (25) is under by approximately 0.6 units? Wait, but let's check the calculation again. Wait, 16 squared is 256, 20 squared is 400. Sum is 656. Square root of 656: let's compute \( \sqrt{656} \). Let's see, 25^2 = 625, 25.6^2 = (25 + 0.6)^2 = 25^2 + 2250.6 + 0.6^2 = 625 + 30 + 0.36 = 655.36. 25.6^2 = 655.36, which is 656 - 655.36 = 0.64 less than 656. So \( \sqrt{656} \approx 25.6 + \frac{0.64}{2*25.6} \) (using linear approximation). The derivative of \( \sqrt{x} \) at x=655.36 is \( \frac{1}{2\sqrt{655.36}} = \frac{1}{51.2} \approx 0.0195 \). So the correction is \( 0.64 * 0.0195 \approx 0.0125 \). So \( \sqrt{656} \approx 25.6 + 0.0125 \approx 25.6125 \). So the actual length is approximately 25.61, and Janice estimated 25. So the difference is 25.61 - 25 = 0.61, so approximately 0.6 units. Since her estimate (25) is less than the actual (25.61), her estimate is under by approximately 0.6 units. Wait, but let's check the options: "It is under by approximately 0.6 units" is an option. Wait, but wait, maybe I mixed up the legs? Wait, maybe 16 and 20 are not both legs? Wait, no, in a right triangle, the hypotenuse is the longest side, so if the other two sides are 16 and 20, the hypotenuse is longer than both, so 16 and 20 are legs. So the calculation is correct.

Answer:

It is under by approximately 0.6 units.