4. the side lengths of different triangles are given determine whether they are right-angled, acute -angled or obtuse-angled
4.1 11; 21; 25
4.2 4√2; 3√2; 5√2
4.3 8; 10; 2
4.1:
To determine whether the given triangle is right-angled, acute-angled, or obtuse-angled, we can use the Pythagorean theorem. According to the theorem, in a right-angled triangle, the sum of the squares of the lengths of the two shorter sides equals the square of the length of the longest side.
In this case, the three side lengths are 11, 21, and 25. Let's find which side is the longest:
11^2 = 121
21^2 = 441
25^2 = 625
Since 625 is the largest square, it corresponds to the longest side of the triangle, which is 25. Now let's check if this triangle is right-angled:
11^2 + 21^2 = 121 + 441 = 562
We can see that 562 is not equal to 625, so this triangle is not right-angled. Since the sum of the squares of the two shorter sides is less than the square of the longest side, this triangle is obtuse-angled.
4.2:
Once again, we can use the Pythagorean theorem to determine the type of triangle. The side lengths in this case are 4√2, 3√2, and 5√2. We need to find which side is the longest by squaring each side length:
(4√2)^2 = 32
(3√2)^2 = 18
(5√2)^2 = 50
Since 50 is the largest square, it corresponds to the longest side of the triangle, which is 5√2. Now let's check if this triangle is right-angled:
(4√2)^2 + (3√2)^2 = 32 + 18 = 50
We can see that 50 is equal to 50, so this triangle is right-angled.
4.3:
For this triangle with side lengths 8, 10, and 2, let's determine the type using the Pythagorean theorem. Squaring each side length gives:
8^2 = 64
10^2 = 100
2^2 = 4
The largest square is 100, corresponding to the longest side length of 10. Now let's check if this triangle is right-angled:
8^2 + 2^2 = 64 + 4 = 68
We can see that 68 is not equal to 100, so this triangle is not right-angled. Since the sum of the squares of the two shorter sides is less than the square of the longest side, this triangle is obtuse-angled.
To determine whether a triangle is right-angled, acute-angled, or obtuse-angled, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's go through each triangle:
4.1 11; 21; 25
To check if it is a right-angled triangle, we can calculate:
11^2 + 21^2 = 121 + 441 = 562
25^2 = 625
Since 625 is equal to 562, we can conclude that this triangle is not a right-angled triangle.
To determine if it is an acute-angled or obtuse-angled triangle, we can compare the squares of the lengths of the sides.
11^2 = 121
21^2 = 441
25^2 = 625
Since 625 is greater than the sum of 121 and 441, we can conclude that this triangle is an obtuse-angled triangle.
4.2 4√2; 3√2; 5√2
To check if it is a right-angled triangle, we can calculate:
(4√2)^2 + (3√2)^2 = 32 + 18 = 50
(5√2)^2 = 50
Since 50 is equal to 50, we can conclude that this triangle is a right-angled triangle.
4.3 8; 10; 2
To check if it is a right-angled triangle, we can calculate:
8^2 + 10^2 = 64 + 100 = 164
2^2 = 4
Since 4 is less than 164, we can conclude that this triangle is an acute-angled triangle.
In summary:
4.1 Triangle with side lengths 11, 21, and 25 is an obtuse-angled triangle.
4.2 Triangle with side lengths 4√2, 3√2, and 5√2 is a right-angled triangle.
4.3 Triangle with side lengths 8, 10, and 2 is an acute-angled triangle.
To determine whether a triangle is right-angled, acute-angled, or obtuse-angled, we need to use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's apply this theorem to the given triangles:
4.1: The side lengths are 11, 21, and 25. To check if it is a right-angled triangle, we can calculate the squares of the side lengths:
11^2 = 121
21^2 = 441
25^2 = 625
If the sum of the squares of the two smaller sides is equal to the square of the largest side, then it is a right-angled triangle. In this case, 121 + 441 = 562, which is not equal to 625. Therefore, triangle 4.1 is not a right-angled triangle.
To determine whether it is acute-angled or obtuse-angled, we can compare the squares of the side lengths:
The sum of the squares of the two smaller sides (121 + 441 = 562) is less than the square of the largest side (625). Therefore, triangle 4.1 is an acute-angled triangle.
4.2: The side lengths are 4√2, 3√2, and 5√2. We can apply the same method to check if it is a right-angled triangle:
(4√2)^2 = 32
(3√2)^2 = 18
(5√2)^2 = 50
The sum of the squares of the two smaller sides (32 + 18 = 50) is equal to the square of the largest side (50). Therefore, triangle 4.2 is a right-angled triangle.
4.3: The side lengths are 8, 10, and 2. Applying the Pythagorean theorem:
8^2 = 64
10^2 = 100
2^2 = 4
The sum of the squares of the two smaller sides (64 + 4 = 68) is not equal to the square of the largest side (100). Therefore, triangle 4.3 is not a right-angled triangle.
To determine whether it is acute-angled or obtuse-angled, we compare the squares of the side lengths:
The sum of the squares of the two smaller sides (64 + 4 = 68) is less than the square of the largest side (100). Therefore, triangle 4.3 is an obtuse-angled triangle.
In summary:
4.1 is an acute-angled triangle.
4.2 is a right-angled triangle.
4.3 is an obtuse-angled triangle.