Exercise 78 Page 659

First, we want to determine whether the given set of numbers can be the measures of the sides of a triangle. To do that we will use the Triangle Inequality Theorem. We have to check if the sum of each two sides is greater than the third side. Note that 6sqrt(3) is about 10.4. 6sqrt(3)+14>& 17 ✓ 6sqrt(3)+17>& 14 ✓ 14+17>& 6sqrt(3) ✓ As we can see, the given side lengths can form a triangle. Now, we need to determine if this triangle is acute, right, or obtuse. To do so, we will compare the square of the largest side length to the sum of the squares of the other two side lengths. Let a, b, and c be the lengths of the sides, with c being the longest.

Condition	Type of Triangle
a^2+b^2 < c^2	Obtuse triangle
a^2+b^2 = c^2	Right triangle
a^2+b^2 > c^2	Acute triangle

Let's now consider the given side lengths 6sqrt(3), 14, and 17. Since 17 is the greatest of the numbers, we will let c be 17. We will also arbitrarily let a be 6sqrt(3) and b be 14. (6sqrt(3))^2+14^2 ? 17^2 Let's simplify the above statement to determine whether the left-hand side is less than, equal to, or greater than the right-hand side.

(6sqrt(3))^2+14^2 ? 17^2

PowProdII

(a b)^m=a^m b^m

6^2(sqrt(3))^2+14^2 ? 17^2

PowSqrt

( sqrt(a) )^2 = a

6^2(3)+14^2 ? 17^2

CalcPow

Calculate power

36(3)+196 ? 289

Multiply

Multiply

108+196 ? 289

AddTerms

Add terms

304 > 289

Referring back to our table, we can conclude that the side lengths 6sqrt(3), 14, and 17 form an acute triangle.