1. Which of the following is the correct equation for the Pythagorean Theorem, where a and b are the side lengths and c is the length of the hypotenuse?
A. (a-b)^2=c^2
B. a^2-b^2=c^2
C. a^2+b^2=c^2 *******
D. (a+b)^2=c^2
2. Given the following information, can you find the coordinates of the unknown point? Explain.
Two points are 14 units apart. The first point is located at (0,7).
A. The unknown point is (12, 0). Substituting into the Pythagorean Theorem gives 14^2=a^2+7^2. Solving this for a gives a value of approximately 12.
B. The unknown point is (0, 21). Since the first point is located on the y-axis, the second point would be located 14 units away from it, also on the y-axis.
C. The unknown point cannot be found. Neither the distance formula nor the Pythagorean Theorem can be sued to work backwards from a distance to find the coordinates of a point. ******
D. The unknown point cannot be found. When substituting into the distance formula, there will be two unknowns, the x-coordinate and the y-coordinate of the unknown point.
3. Find the distance between the points (9,−7) and (5,−4).
A. 25
B. √7
C. 5 *******
D. √137
4. To find the distance between (17,3) and (17,−5), Marcia used the following equation. Is Marcia correct? Explain.
D = | 3 - (- 5) | = 8
A. Marcia is not correct. According to the distance formula, the distance should be
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√ (-17-17)^2+(3-(-5))^2=√8 (I hope you understand what I tried to do here) **********
B. Marcia is correct. Since the x-coordinates are the same, the distance between the points can be treated as if they are in one-dimension.
C. Marcia is not correct. Since the points are in two-dimensions, the distance formula must be used to find the distance.
D. Marcia is correct. For any pair of points, the distance between the points can be treated as if they are in one-dimension.
5. Which of the following is the correct formula for distance?
A. d=
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√ x2-x1)+(y2-y1), where d is the distance between points (x1,y1) and (x2,y2). *******
B. d=
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√ x2-x1)^2 +(y2-y1)^2, where d is the distance between points (x1,y1) and (x2,y2).
C. d= (x2−x1)^2+(y2−y1)^2, where d is the distance between points (x1,y1) and (x2,y2).
D. d=
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√ x2+x1)^2+(y2+y1)^2, where d is the distance between points (x1,y1) and (x2,y2).
If someone could check over these, maybe even just a couple of them, maybe even just one of them, it would be so greatly appreciated. I'm having a really tough time. Thank you so much
# 1,2,3 ok
4 B
5 B allowing for the missing parentheses
1. C. a^2+b^2=c^2 is the correct equation for the Pythagorean Theorem, where a and b are the side lengths and c is the length of the hypotenuse.
2. C. The unknown point cannot be found. Neither the distance formula nor the Pythagorean Theorem can be used to work backward from a distance to find the coordinates of a point. In this case, we only have the distance between the points and the y-coordinate of one point, which is insufficient information to determine the coordinates of the unknown point.
3. C. The distance between the points (9,-7) and (5,-4) can be found using the distance formula:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
d = √[(5 - 9)^2 + (-4 - (-7))^2]
d = √[(-4)^2 + 3^2]
d = √[16 + 9]
d = √25
d = 5
Therefore, the distance between the points is 5.
4. A. Marcia is not correct. According to the distance formula, the distance between (17,3) and (17,-5) should be:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
d = √[(17 - 17)^2 + (3 - (-5))^2]
d = √[0 + 8^2]
d = √[64]
d = 8.
So, the correct distance between the points is 8.
5. B. The correct formula for distance is:
d = √[(x2 - x1)^2 + (y2 - y1)^2], where d is the distance between points (x1, y1) and (x2, y2).
1. The correct equation for the Pythagorean Theorem is C. a^2+b^2=c^2. This equation 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 (a and b).
2. The correct answer is C. The unknown point cannot be found. Neither the distance formula nor the Pythagorean Theorem can be used to work backwards from a distance to find the coordinates of a point. These formulas are used to find the distance between given points, not to find the coordinates of an unknown point given a distance.
3. To find the distance between two points, you can use the distance formula. The distance formula is given by the equation:
d = √[(x2-x1)^2 + (y2-y1)^2]
Using the coordinates (9, -7) and (5, -4), we can substitute these values into the distance formula:
d = √[(5-9)^2 + (-4-(-7))^2]
= √[(-4)^2 + (3)^2]
= √[16 + 9]
= √25
= 5
Therefore, the distance between the two points is C. 5.
4. Marcia is correct. Since the points (17,3) and (17,-5) have the same x-coordinate, their distance in the y-direction can be found simply by taking the absolute difference between their y-coordinates, which is 3 - (-5) = 8. Therefore, the correct answer is A. Marcia is correct.
5. The correct formula for distance is B. d = √[(x2-x1)^2 + (y2-y1)^2], where d is the distance between points (x1,y1) and (x2,y2). This formula is also known as the distance formula in 2-dimensional coordinate geometry. It calculates the straight-line distance between two points in a coordinate plane by finding the differences in the x-coordinates and y-coordinates, squaring these differences, taking the sum, and then taking the square root of the sum to obtain the distance.