Question
Is the student correct?
Student A was given the three sides of a triangle 3, 4, 5 and plugged in the values to the pythagorean theorem. Student A concluded the triangle is a right triangle.
Question 1 options:
True
False
Question 2 (2 points)
The pythagorean theorem can only be used with right triangles.
Question 2 options:
True
False
Question 3 (4 points)
Which of the following sets of triangle length will form a right triangle?
Select 4 correct answer(s)
Question 3 options:
3, 4, 5
3, 7, 10
6, 8, 10
5, 12, 13
10, 24, 26
20, 30, 40
Question 4 (1 point)
Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point?
Question 4 options:
9.4
7.4
4.4
4.9
Student A was given the three sides of a triangle 3, 4, 5 and plugged in the values to the pythagorean theorem. Student A concluded the triangle is a right triangle.
Question 1 options:
True
False
Question 2 (2 points)
The pythagorean theorem can only be used with right triangles.
Question 2 options:
True
False
Question 3 (4 points)
Which of the following sets of triangle length will form a right triangle?
Select 4 correct answer(s)
Question 3 options:
3, 4, 5
3, 7, 10
6, 8, 10
5, 12, 13
10, 24, 26
20, 30, 40
Question 4 (1 point)
Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point?
Question 4 options:
9.4
7.4
4.4
4.9
Answers
GPT-4o mini
Let's analyze each question one by one.
**Question 1: Is the student correct?**
Student A was given the three sides of a triangle 3, 4, 5 and plugged in the values to the Pythagorean theorem. Student A concluded the triangle is a right triangle.
To check if a triangle with sides 3, 4, and 5 is a right triangle, we can use the Pythagorean theorem (a² + b² = c²), where c is the longest side.
Here, the longest side (hypotenuse) is 5, and the other two sides are 3 and 4.
Calculating:
3² + 4² = 9 + 16 = 25
5² = 25
Since both sides of the equation are equal, 3, 4, and 5 do indeed form a right triangle.
**Answer for Question 1: True**
---
**Question 2: The Pythagorean theorem can only be used with right triangles.**
The Pythagorean theorem specifically applies to right triangles. It's used to relate the lengths of the sides in a right triangle. For non-right triangles, this theorem does not apply.
**Answer for Question 2: True**
---
**Question 3: Which of the following sets of triangle lengths will form a right triangle?**
We'll check each set of lengths against the Pythagorean theorem:
1. **3, 4, 5:** (3² + 4² = 5²) -> 9 + 16 = 25 -> **True**
2. **3, 7, 10:** (3² + 7² = 10²) -> 9 + 49 = 58 (10² = 100) -> **False**
3. **6, 8, 10:** (6² + 8² = 10²) -> 36 + 64 = 100 -> **True**
4. **5, 12, 13:** (5² + 12² = 13²) -> 25 + 144 = 169 -> **True**
5. **10, 24, 26:** (10² + 24² = 26²) -> 100 + 576 = 676 (26² = 676) -> **True**
6. **20, 30, 40:** (20² + 30² = 40²) -> 400 + 900 = 1300 (40² = 1600) -> **False**
The sets of lengths that form a right triangle are:
- 3, 4, 5
- 6, 8, 10
- 5, 12, 13
- 10, 24, 26
**Answer for Question 3:**
3, 4, 5; 6, 8, 10; 5, 12, 13; 10, 24, 26
---
**Question 4: Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point?**
To find the shortest distance between their final point and the starting point, we can visualize this as a right triangle where:
- One leg is 8 miles (north)
- The other leg is 5 miles (west)
We can use the Pythagorean theorem to find the hypotenuse:
c = √(8² + 5²) = √(64 + 25) = √89
Calculating √89 gives approximately:
c ≈ 9.4 miles.
**Answer for Question 4: 9.4**
**Question 1: Is the student correct?**
Student A was given the three sides of a triangle 3, 4, 5 and plugged in the values to the Pythagorean theorem. Student A concluded the triangle is a right triangle.
To check if a triangle with sides 3, 4, and 5 is a right triangle, we can use the Pythagorean theorem (a² + b² = c²), where c is the longest side.
Here, the longest side (hypotenuse) is 5, and the other two sides are 3 and 4.
Calculating:
3² + 4² = 9 + 16 = 25
5² = 25
Since both sides of the equation are equal, 3, 4, and 5 do indeed form a right triangle.
**Answer for Question 1: True**
---
**Question 2: The Pythagorean theorem can only be used with right triangles.**
The Pythagorean theorem specifically applies to right triangles. It's used to relate the lengths of the sides in a right triangle. For non-right triangles, this theorem does not apply.
**Answer for Question 2: True**
---
**Question 3: Which of the following sets of triangle lengths will form a right triangle?**
We'll check each set of lengths against the Pythagorean theorem:
1. **3, 4, 5:** (3² + 4² = 5²) -> 9 + 16 = 25 -> **True**
2. **3, 7, 10:** (3² + 7² = 10²) -> 9 + 49 = 58 (10² = 100) -> **False**
3. **6, 8, 10:** (6² + 8² = 10²) -> 36 + 64 = 100 -> **True**
4. **5, 12, 13:** (5² + 12² = 13²) -> 25 + 144 = 169 -> **True**
5. **10, 24, 26:** (10² + 24² = 26²) -> 100 + 576 = 676 (26² = 676) -> **True**
6. **20, 30, 40:** (20² + 30² = 40²) -> 400 + 900 = 1300 (40² = 1600) -> **False**
The sets of lengths that form a right triangle are:
- 3, 4, 5
- 6, 8, 10
- 5, 12, 13
- 10, 24, 26
**Answer for Question 3:**
3, 4, 5; 6, 8, 10; 5, 12, 13; 10, 24, 26
---
**Question 4: Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point?**
To find the shortest distance between their final point and the starting point, we can visualize this as a right triangle where:
- One leg is 8 miles (north)
- The other leg is 5 miles (west)
We can use the Pythagorean theorem to find the hypotenuse:
c = √(8² + 5²) = √(64 + 25) = √89
Calculating √89 gives approximately:
c ≈ 9.4 miles.
**Answer for Question 4: 9.4**