a Suppose that the side lengths of a triangle are a, b, and c, with c being the longest. If a^2+b^2>c^2, the triangle is an acute triangle.
B
b Suppose that the side lengths of a triangle are a, b, and c, with c being the longest. If a^2+b^2obtuse triangle.
A
aExample Integer: 10
B
bExample Integer: 11
Practice makes perfect
a Suppose that the side lengths of a triangle are a, b, and c, with c being the longest. If we compare a^2+b^2 to c^2, we can find the type of triangle these sides form.
The side lengths 6, 9, and sqrt(117)≈ 10.8 form a right triangle. Therefore, for 6, 9, and the integer number j to be the sides of an acute triangle, j must be greater than 6 and 9, but less than sqrt(117)≈ 10.8.
6 < 9 < j < 10.8
A number that satisfies these conditions is j=10. Let's verify that 6^2+9^2 is greater than 10^2.
Please note that we could have also assumed that 9 was the longest side and solved for a, so our answer is only one possible correct answer.
b In Part A we found that the given two side lengths, 6 and 9, and the number sqrt(117)≈ 10.8 form a right triangle. Therefore, for 6, 9, and the integer k to represent the side lengths of an obtuse triangle, k must be greater than sqrt(117)≈ 10.8.
6 < 9 < 10.8 < k
A number that satisfies this condition is k=11. Let's verify that 6^2+9^2 is less than 11^2.
