Apply the Triangle Inequality Theorem to determine if sides with lengths of 3 feet, 4 feet, and 5 feet form a triangle.(1 point) Responses Yes, because the sum of any two sides is less than the third side. Yes, because the sum of any two sides is less than the third side. No, because the sum of any two sides is less than the third side. No, because the sum of any two sides is less than the third side. Yes, because the sum of any two sides is greater than the third side. Yes, because the sum of any two sides is greater than the third side. No, because two out of three conditions are met. No, because two out of three conditions are met.
The Triangle Inequality Theorem states that, for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. We can test this theorem on the given set of side lengths: 3 feet, 4 feet, and 5 feet.
Let's check each pair of sides:
1. 3 + 4 > 5 (7 > 5) - True
2. 3 + 5 > 4 (8 > 4) - True
3. 4 + 5 > 3 (9 > 3) - True
Since the sum of the lengths of any two sides is greater than the length of the remaining side in all three cases, a triangle can indeed be formed with these side lengths.
The correct response is: Yes, because the sum of any two sides is greater than the third side.
If a right triangle has an angle that is 65° , what is the third angle in the triangle?(1 point) $$
In a right triangle, one angle is always 90° because it is a right angle. Let's denote the given angle as A, the right angle as B, and the unknown angle as C.
Since the angles in a triangle always add up to 180°, we can set up the following equation to solve for the unknown angle C:
A + B + C = 180°
Given that angle A is 65° and angle B is 90° (the right angle), we have:
65° + 90° + C = 180°
155° + C = 180°
Now, subtract 155° from both sides to find the measure of angle C:
C = 180° - 155°
C = 25°
Therefore, the third angle in the triangle is 25°.
Use the image to answer the question. Explain a proof of the Pythagorean Theorem using the image. (4 points) Math Short Answer Rubric (4 points) Points Awarded Criteria 4 The answer is clear and correct. The answer shows that the student used problem-solving skills and reasoning. All work is shown or explained. 3 The answer is mostly correct. The student made small errors while solving. The answer shows that the student used some problem-solving skills and reasoning. Most work is shown or explained. 2 The answer is partially correct. The student made errors while solving. The answer shows that the student used some problem-solving skills or reasoning. Some work is shown or explained. 1 The answer is incorrect. The student made errors while solving. The answer shows little problem-solving skills or reasoning. Little or no work is shown. 0 The question is not answered. Skip to navigation
Unfortunately, I cannot view the image you're referring to provide a precise proof of the Pythagorean Theorem based on that image. However, I can give you a general description of a common visual proof of the Pythagorean Theorem.
One of the most famous visual proofs uses a square divided into two smaller squares and two identical rectangles. Here is a general explanation assuming you have an image following this description:
1. Imagine a large square whose sides are each (a + b) units long, where 'a' and 'b' are the lengths of the two legs of a right triangle.
2. Inside this square are drawn four identical right triangles, each with the legs measuring 'a' and 'b' and the hypotenuse 'c'. These triangles are arranged in such a way that their hypotenuses form a smaller square in the center of the larger square.
3. The area of the large square is (a + b)².
4. The area of the large square can also be broken down into the areas of the four triangles and the smaller square. The four right triangles together have an area of 4 * (1/2) * a * b = 2ab. The smaller square in the center, which has sides of length 'c', has an area of c².
5. By setting the two expressions for the area of the large square equal to each other, we get:
(a + b)² = 4 * (1/2) * a * b + c²
a² + 2ab + b² = 2ab + c²
6. When you subtract the 2ab from both sides, you're left with the Pythagorean Theorem:
a² + b² = c²
This visual proof relies on the rearrangement of the four identical triangles and the remaining central smaller square to show that the sum of the areas of the squares on the triangle's legs is equal to the area of the square on its hypotenuse. If you had a specific image in mind, please provide more details or describe the image, and I can tailor the explanation to that particular visual proof.
To apply the Triangle Inequality Theorem, we need to determine if the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
In this case, we have sides with lengths of 3 feet, 4 feet, and 5 feet. Let's check if the sum of any two sides is greater than the third side:
1. Sum of 3 feet and 4 feet is 7 feet, which is greater than 5 feet.
2. Sum of 3 feet and 5 feet is 8 feet, which is greater than 4 feet.
3. Sum of 4 feet and 5 feet is 9 feet, which is greater than 3 feet.
Since the sum of any two sides is greater than the third side, the given lengths of 3 feet, 4 feet, and 5 feet form a triangle.
Therefore, the correct response is: "Yes, because the sum of any two sides is greater than the third side."