Given the measures of all three sides of a triangle, we can determine whether it is right, acute, or obtuse by using the Converse of the Pythagorean Theorem and Pythagorean Inequalities Theorems. In each case, c is the length of the longest side.
Right: If a^2+ b^2= c^2, then the triangle is a right triangle.
Acute: If c^2 < a^2 + b^2, then the triangle is an acute triangle.
Obtuse: If c^2 > a^2 + b^2, then the triangle is an obtuse triangle.
Let's begin by plotting the given vertices on a coordinate plane and drawing the triangle.
Now, we can find the measure of each side using the Distance Formula.
Side
Distance Formula
Simplify
XY, ( 1,2), ( 4,6)
sqrt(( 4- 1)^2+( 6- 2)^2)
5
YZ, ( 4,6), (6,6)
sqrt((6- 4)^2+(6- 6)^2)
2
ZX, (6,6), ( 1,2)
sqrt(( 1-6)^2+( 2-6)^2)
sqrt(41)
In this case, sqrt(41) is the length of the longest side. Let's check whether the equation, or one of the inequalities, is satisfied.
We can tell that the left-hand side is greater than the right-hand side. Therefore, the sign that will satisfy this relation is >. Now we can rewrite the original inequality with this sign.
( sqrt(41))^2 > 2^2 + 5^2
As we can see, the lengths of the sides of our triangle satisfy c^2 > a^2 + b^2. Therefore, it is an obtuse triangle.
