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: 11
B
bExample Integer: 12
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 5, 10, and sqrt(125)≈ 11.2 form a right triangle. Therefore, for 5, 10, and the integer number j to be the sides of an acute triangle, j must be greater than 5 and 10, but less than sqrt(125)≈ 11.2.
5 < 10 < j < 11.2
A number that satisfies these conditions is j=11. Let's verify that 5^2+10^2 is greater than 11^2.
Please note that we could have also assumed that 10 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, 5 and 10, and the number sqrt(125)≈ 11.2 form a right triangle. Therefore, for 5, 10, and the integer k to represent the side lengths of an obtuse triangle, k must be greater than sqrt(125)≈ 11.2.
5 < 10 < 11.2 < k
A number that satisfies this condition is k=12. Let's verify that 5^2+10^2 is less than 12^2.
