I need to find out the type of triangle (by the type of angles). Is triangle acute, right or obtuse, when the lenghts of triangle sides are:
a) 7, 8, 12
b) 20, 15, 40
c) 2/3, 8/9, 10/9
The largest angle is opposite the largest side.
So all we have to do is check if the largest side is greater or smaller than a "hypotenuse"
in 1st:
7^2 + 8^2 = 113
12^2 = 144
so 12^2 > 7^2 + 8^2
and the triangle is obtuse with the angle opposite 12 as > 90°
2nd:
a trick question!
To form a triangle, the sum of any two sides has to be greater than the third side,
but 20+15 is NOT greater than 40.
So we can't even form the triangle.
3rd:
2/3, 8/9, 10/9
or
6/9, 8/9, 10/9
this triangle has the same angles as
6 , 8, 9 (they are similar)
check for 9^2 vs 6^2 + 8^2
To determine the type of triangle based on the angles, you need to use the Pythagorean theorem and the concept of the relationship between the squares of the sides.
Here's how you can find the answer for each of the given triangles:
a) Triangle with side lengths 7, 8, 12:
To determine if this triangle is acute, right, or obtuse, we need to check the relationship between the squares of the sides.
Step 1: Square each side length.
7^2 = 49
8^2 = 64
12^2 = 144
Step 2: Arrange the squared side lengths in ascending order.
49, 64, 144
Step 3: Check the relationship between the squared side lengths.
Since the sum of the two smallest squared side lengths (49 + 64 = 113) is less than the squared length of the longest side (144), the triangle is acute.
Therefore, the triangle with side lengths 7, 8, 12 is an acute triangle.
b) Triangle with side lengths 20, 15, 40:
Again, we need to determine if this triangle is acute, right, or obtuse based on the relationship between the squares of the sides.
Step 1: Square each side length.
20^2 = 400
15^2 = 225
40^2 = 1600
Step 2: Arrange the squared side lengths in ascending order.
225, 400, 1600
Step 3: Check the relationship between the squared side lengths.
Since the sum of the two smallest squared side lengths (225 + 400 = 625) is equal to the squared length of the longest side (1600), the triangle is right.
Therefore, the triangle with side lengths 20, 15, 40 is a right triangle.
c) Triangle with side lengths 2/3, 8/9, 10/9:
To determine the triangle's type, we'll apply the same steps as before.
Step 1: Square each side length.
(2/3)^2 = 4/9
(8/9)^2 = 64/81
(10/9)^2 = 100/81
Step 2: Arrange the squared side lengths in ascending order.
4/9, 64/81, 100/81
Step 3: Check the relationship between the squared side lengths.
Since the sum of the two smallest squared side lengths (4/9 + 64/81 = 424/729) is greater than the squared length of the longest side (100/81), the triangle is obtuse.
Therefore, the triangle with side lengths 2/3, 8/9, 10/9 is an obtuse triangle.
To summarize:
a) Triangle with side lengths 7, 8, 12 is an acute triangle.
b) Triangle with side lengths 20, 15, 40 is a right triangle.
c) Triangle with side lengths 2/3, 8/9, 10/9 is an obtuse triangle.