Question

applying the converse of the pythagorean theorem
janice examines the given triangle and estimates that
the longest side has a length of 25 units—if it is a right
triangle. how does her estimate compare to the actual
length?
o it is exactly correct.
o it is under by approximately 0.6 units.
o it is over by approximately 0.6 units.
o it is over by 13 units.

Explanation:

Step1: Apply the Pythagorean theorem

If the triangle is a right - triangle, let \(a = 16\), \(b=20\), and \(c\) be the length of the hypotenuse. According to the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\).
Substitute \(a = 16\) and \(b = 20\) into the formula: \(c=\sqrt{16^{2}+20^{2}}=\sqrt{256 + 400}=\sqrt{656}\).

Step2: Calculate the value of \(\sqrt{656}\)

We know that \(\sqrt{656}\approx25.6\) (since \(25^{2}=625\) and \(26^{2}=676\), and \(656-625 = 31\), \(676 - 656=20\), using linear approximation or a calculator \(\sqrt{656}\approx25.6\)).

Step3: Compare the estimated value and the actual value

The estimated value is \(25\), and the actual value is approximately \(25.6\). The difference is \(25.6-25=0.6\).

Answer:

It is under by approximately \(0.6\) units.