Question

persevere with problems find the missing measure in each figure below. round to the nearest tenth if necessary. 16. 17. lesson 6 use the pythagorean theorem

Explanation:

Step1: Find the diagonal of the base

For problem 16:
The base is a rectangle with length \(l = 12\) mm and width \(w=5\) mm.
By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 12\), \(b = 5\) and \(c\) is the diagonal of the base.
\(c=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\) mm.

Step2: Find the space - diagonal

Now, consider a right - triangle formed by the diagonal of the base (\(13\) mm), the height of the prism (\(h = 5\) mm) and the space - diagonal \(x\).
Again, using the Pythagorean theorem \(x=\sqrt{13^{2}+5^{2}}=\sqrt{169+25}=\sqrt{194}\approx13.9\) mm.

For problem 17:

Step1: Find the diagonal of the base

The base is a square with side \(s = 10\) in.
By the Pythagorean theorem, the diagonal of the base \(d=\sqrt{10^{2}+10^{2}}=\sqrt{100 + 100}=\sqrt{200}=10\sqrt{2}\approx14.1\) in.

Step2: Find the space - diagonal

Consider a right - triangle formed by the diagonal of the base (\(d\approx14.1\) in), the height of the prism (\(h = 15\) in) and the space - diagonal \(x\).
Using the Pythagorean theorem \(x=\sqrt{(10\sqrt{2})^{2}+15^{2}}=\sqrt{200+225}=\sqrt{425}\approx20.6\) in.

Answer:

1. \(x\approx13.9\) mm; 17. \(x\approx20.6\) in.