Question

1. the diagonal of a rectangle is 25 in. the width is 15 inches. what is the length? 6. two sides of a right triangle are 8 and 12. find the hypotenuse.

Explanation:

Step1: Recall Pythagorean theorem for right - triangle

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\). In problem 6, \(a = 8\) and \(b = 12\).
\[c^{2}=8^{2}+12^{2}\]
\[c^{2}=64 + 144\]
\[c^{2}=208\]
\[c=\sqrt{208}=4\sqrt{13}\approx 14.42\]

Step2: Recall Pythagorean theorem for rectangle

In a rectangle, the diagonal \(d\), length \(l\) and width \(w\) satisfy \(d^{2}=l^{2}+w^{2}\). In problem 5, \(d = 25\) and \(w = 15\). Let the length be \(l\). Then \(l^{2}=d^{2}-w^{2}\).
\[l^{2}=25^{2}-15^{2}\]
\[l^{2}=(25 + 15)(25 - 15)\] (using \(a^{2}-b^{2}=(a + b)(a - b)\))
\[l^{2}=40\times10\]
\[l^{2}=400\]
\[l = 20\]

Answer:

1. 20 inches
2. \(4\sqrt{13}\approx14.42\)