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
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
Example Integer: 4
B
Example Integer: 5
Practice makes perfect
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 2, 4, and sqrt(20)≈ 4.47 form a right triangle. Therefore, for 2, 4, and the integer number j to be the sides of an acute triangle, j must be greater than or equal to 2 and 4, but less than sqrt(20)≈ 4.47.
2 < 4 ≤ j < 4.47
A number that satisfies these conditions is j=4. Let's verify that 2^2+4^2 is greater than 4^2.
Please note that we could have also assumed that 4 was the longest side and solved for a, so our answer is only one possible correct answer.
In Part A we found that the given two side lengths, 2 and 4, and the number sqrt(20)≈ 4.47 form a right triangle. Therefore, for 2, 4, and the integer k to represent the side lengths of an obtuse triangle, k must be greater than sqrt(20)≈ 4.47.
2 < 4 < 4.47 < k
A number that satisfies this condition is k=5. Let's verify that 2^2+4^2 is less than 5^2.
