Question

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? 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.

Explanation:

Step1: Apply Pythagorean theorem

Let the two given sides be \(a = 16\) and \(b=20\). For a right - triangle, if the two legs are \(a\) and \(b\), the hypotenuse \(c\) is given by \(c=\sqrt{a^{2}+b^{2}}\). But if the hypotenuse is \(b = 20\) and one leg is \(a = 16\), then the other leg \(x=\sqrt{b^{2}-a^{2}}\). So \(x=\sqrt{20^{2}-16^{2}}=\sqrt{(20 + 16)(20 - 16)}=\sqrt{36\times4}=\sqrt{144}=12\). If we assume the two legs are \(a = 16\) and \(b = 12\), the hypotenuse \(c=\sqrt{16^{2}+12^{2}}=\sqrt{256 + 144}=\sqrt{400}=20\).

Step2: Compare with the estimate

The estimate is 25 units. The actual length of the longest side (hypotenuse) is 20 units. The difference is \(25-20 = 5\) units. Since \(5\approx0.6\times8.33\) (approximate value), the estimate is over by approximately \(0.6\) units.

Answer:

It is over by approximately 0.6 units.