We can solve this problem using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides.
In this case, one side has a length of 6 ft and the other side has a length of 8 ft. So, we have:
c^2 = 6^2 + 8^2
c^2 = 36 + 64
c^2 = 100
Taking the square root of both sides, we have:
c = √100
c = 10
Therefore, the length of the hypotenuse (side c) is 10 ft.
What is the length of the hypotenuse? If necessary, round to the nearest tenth.
image shows a right triangle with a side of 6 and a side of 8 ft with another side being labeled c
11 answers
What is the length of the missing leg? If necessary, round to the nearest 10th.
image shows a triangle with a side of 20 cm, another side with the length of 12 cm, and another side labeled b
image shows a triangle with a side of 20 cm, another side with the length of 12 cm, and another side labeled b
To find the length of the missing leg in this right triangle, we can again use the Pythagorean theorem.
One side has a length of 20 cm and the other side has a length of 12 cm. Let's label the missing leg as b.
Using the Pythagorean theorem, we have:
b^2 = 20^2 - 12^2
b^2 = 400 - 144
b^2 = 256
Taking the square root of both sides, we have:
b = √256
b = 16
Therefore, the length of the missing leg (side b) is 16 cm.
One side has a length of 20 cm and the other side has a length of 12 cm. Let's label the missing leg as b.
Using the Pythagorean theorem, we have:
b^2 = 20^2 - 12^2
b^2 = 400 - 144
b^2 = 256
Taking the square root of both sides, we have:
b = √256
b = 16
Therefore, the length of the missing leg (side b) is 16 cm.
Laya correctly used the diagram shown to explain the Pythagorean theorem to a classmate. Which statement did Laya use in her explanation?
A) 5^2 + 5^2 = 3^2
B) 3^2 + 4^2 = 5^2
C) 5^2 + 5^2 < 3^2
D) 3^2 + 4^2 > 5^2
Image shows 3 squares. One with the area of 25, another within area of 16, and another with an area of 9.
A) 5^2 + 5^2 = 3^2
B) 3^2 + 4^2 = 5^2
C) 5^2 + 5^2 < 3^2
D) 3^2 + 4^2 > 5^2
Image shows 3 squares. One with the area of 25, another within area of 16, and another with an area of 9.
Laya correctly used the Pythagorean theorem to explain to her classmate. The statement she used is:
B) 3^2 + 4^2 = 5^2
This statement correctly represents the Pythagorean theorem, which states that in a right 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. In this case, Laya used the lengths 3, 4, and 5 as an example to show that the squares of the lengths of the two shorter sides (3^2 and 4^2) add up to the square of the length of the hypotenuse (5^2).
B) 3^2 + 4^2 = 5^2
This statement correctly represents the Pythagorean theorem, which states that in a right 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. In this case, Laya used the lengths 3, 4, and 5 as an example to show that the squares of the lengths of the two shorter sides (3^2 and 4^2) add up to the square of the length of the hypotenuse (5^2).
The following side lengths, in meters, were given to a carpenter to build a front porch with a triangular design. The carpenter needs to determine which set of lengths will make a triangle to be able to use it in his design. Option 1: Side lengths: 3, 3, 6 Option 2: Side lengths: 6, 8, 10 Option 3: Side lengths: 4, 6, 13
Responses
Option 1
Option 2
Option 3
Responses
Option 1
Option 2
Option 3
To determine if a set of side lengths can form a triangle, we need to apply the triangle inequality theorem. According to this theorem, for a triangle with side lengths a, b, and c:
a + b > c
a + c > b
b + c > a
Let's evaluate each option:
Option 1: Side lengths: 3, 3, 6
3 + 3 > 6 (6 > 6) ✘
3 + 6 > 3 (9 > 3) ✔
3 + 6 > 3 (9 > 3) ✔
Option 1 does not satisfy the triangle inequality theorem, so it cannot form a triangle.
Option 2: Side lengths: 6, 8, 10
6 + 8 > 10 (14 > 10) ✔
6 + 10 > 8 (16 > 8) ✔
8 + 10 > 6 (18 > 6) ✔
Option 2 satisfies the triangle inequality theorem, so it can form a triangle.
Option 3: Side lengths: 4, 6, 13
4 + 6 > 13 (10 > 13) ✘
4 + 13 > 6 (17 > 6) ✔
6 + 13 > 4 (19 > 4) ✔
Option 3 does not satisfy the triangle inequality theorem, so it cannot form a triangle.
Based on the evaluation, the carpenter can use Option 2 (side lengths of 6, 8, and 10) to construct a triangle for his front porch design.
a + b > c
a + c > b
b + c > a
Let's evaluate each option:
Option 1: Side lengths: 3, 3, 6
3 + 3 > 6 (6 > 6) ✘
3 + 6 > 3 (9 > 3) ✔
3 + 6 > 3 (9 > 3) ✔
Option 1 does not satisfy the triangle inequality theorem, so it cannot form a triangle.
Option 2: Side lengths: 6, 8, 10
6 + 8 > 10 (14 > 10) ✔
6 + 10 > 8 (16 > 8) ✔
8 + 10 > 6 (18 > 6) ✔
Option 2 satisfies the triangle inequality theorem, so it can form a triangle.
Option 3: Side lengths: 4, 6, 13
4 + 6 > 13 (10 > 13) ✘
4 + 13 > 6 (17 > 6) ✔
6 + 13 > 4 (19 > 4) ✔
Option 3 does not satisfy the triangle inequality theorem, so it cannot form a triangle.
Based on the evaluation, the carpenter can use Option 2 (side lengths of 6, 8, and 10) to construct a triangle for his front porch design.
Determine if each of the following statements is always true, sometimes true or never true. A. The sum of the measures of two supplementary angles is 180°. B. The sum of complimentary angles is 180°. C. Two adjacent angles are complementary.
A. The sum of the measures of two supplementary angles is 180°.
This statement is always true. By definition, supplementary angles are a pair of angles that add up to 180°.
B. The sum of complimentary angles is 180°.
This statement is never true. By definition, complementary angles are a pair of angles that add up to 90°, not 180°.
C. Two adjacent angles are complementary.
This statement is sometimes true. Adjacent angles are angles that share a common vertex and a common side, but they may or may not be complementary. Complementary angles add up to 90°, so if the measures of two adjacent angles add up to 90°, then they are complementary. However, if the measures do not add up to 90°, then they are not complementary.
This statement is always true. By definition, supplementary angles are a pair of angles that add up to 180°.
B. The sum of complimentary angles is 180°.
This statement is never true. By definition, complementary angles are a pair of angles that add up to 90°, not 180°.
C. Two adjacent angles are complementary.
This statement is sometimes true. Adjacent angles are angles that share a common vertex and a common side, but they may or may not be complementary. Complementary angles add up to 90°, so if the measures of two adjacent angles add up to 90°, then they are complementary. However, if the measures do not add up to 90°, then they are not complementary.
which r true which r false
Looking at the statements again:
A. The sum of the measures of two supplementary angles is 180°.
This statement is true.
B. The sum of complimentary angles is 180°.
This statement is false.
C. Two adjacent angles are complementary.
This statement is sometimes true and sometimes false, depending on whether the measures of the adjacent angles add up to 90° (complementary) or not.
To summarize:
A. is true.
B. is false.
C. is sometimes true and sometimes false.
A. The sum of the measures of two supplementary angles is 180°.
This statement is true.
B. The sum of complimentary angles is 180°.
This statement is false.
C. Two adjacent angles are complementary.
This statement is sometimes true and sometimes false, depending on whether the measures of the adjacent angles add up to 90° (complementary) or not.
To summarize:
A. is true.
B. is false.
C. is sometimes true and sometimes false.
7 answers