Question

distance formula: part 1
1 more question to go
which of the following would be an incorrect way to substitute the points (2,3) and (-4, -3) into the distance formula?
○ $\sqrt{(-4 - 2)^2 + (3 - 3)^2}$
○ $\sqrt{(2 - 4)^2 + (3 - 3)^2}$
○ $\sqrt{(2 - (-4))^2 + (3 - (-3))^2}$
○ two of these are incorrect.
submit

Explanation:

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) (or equivalently \(\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}\) since squaring eliminates the sign difference).

Step 1: Analyze the first option

For points \((2,3)\) (let \(x_1 = 2,y_1 = 3\)) and \((- 4,-3)\) (let \(x_2=-4,y_2 = - 3\)), the first option is \(\sqrt{(-4 - 2)^2+(3 - 3)^2}\). Here, \(x_2 - x_1=-4 - 2\) and \(y_2 - y_1=3 - 3\) (wait, \(y_2=-3\), so \(y_2 - y_1=-3 - 3=-6\), but in the first option \(y_2 - y_1\) is written as \(3 - 3\) which is wrong. Wait, no, maybe we swapped \(x_1,y_1\) and \(x_2,y_2\). If we take \((x_1=-4,y_1 = - 3)\) and \((x_2 = 2,y_2=3)\), then \(x_2 - x_1=2-(-4)=6\), \(y_2 - y_1=3-(-3) = 6\). But the first option has \(x_2 - x_1=-4 - 2=-6\) (squaring will make it \(36\)) and \(y_2 - y_1=3 - 3 = 0\) (which is wrong because \(y_2 - y_1\) should be \(3-(-3)=6\) or \(-3 - 3=-6\)). Wait, no, let's re - evaluate.

Wait, the first option: \(\sqrt{(-4 - 2)^2+(3 - 3)^2}\). Let's check the \(y\) - component. The \(y\) - coordinates of the points are \(3\) and \(-3\). So \(y_2 - y_1\) should be either \(3-(-3)=6\) or \(-3 - 3=-6\), but in the first option, it is \(3 - 3 = 0\), which is incorrect.

Step 2: Analyze the second option

The second option is \(\sqrt{(2 - 4)^2+(3 - 3)^2}\). The \(x\) - coordinates of the given points are \(2\) and \(-4\), not \(2\) and \(4\). So the \(x\) - component \(2 - 4\) is incorrect (it should be \(2-(-4)\) or \(-4 - 2\)) and the \(y\) - component \(3 - 3\) is also incorrect (should be \(3-(-3)\) or \(-3 - 3\)).

Step 3: Analyze the third option

The third option is \(\sqrt{(2-(-4))^2+(3 - (-3))^2}\). Using the distance formula with \((x_1 = 2,y_1 = 3)\) and \((x_2=-4,y_2=-3)\), \(x_2 - x_1=-4 - 2=-6\), \(x_1 - x_2=2-(-4)=6\) (since \((x_1 - x_2)^2=(x_2 - x_1)^2\)) and \(y_2 - y_1=-3 - 3=-6\), \(y_1 - y_2=3-(-3)=6\) (since \((y_1 - y_2)^2=(y_2 - y_1)^2\)). So this substitution is correct.

Step 4: Analyze the fourth option

Since the first and the second options are incorrect, the statement "Two of these are incorrect" is correct? Wait, no, wait. Wait, the first option: \(\sqrt{(-4 - 2)^2+(3 - 3)^2}\). Let's recalculate the \(y\) - difference. The two points are \((2,3)\) and \((-4,-3)\). The \(y\) - difference is \(3-(-3)=6\) or \(-3 - 3=-6\). In the first option, the \(y\) - difference is \(3 - 3 = 0\), which is wrong. The second option: \(\sqrt{(2 - 4)^2+(3 - 3)^2}\). The \(x\) - difference should be \(2-(-4)\) or \(-4 - 2\), not \(2 - 4\), and the \(y\) - difference is wrong. The third option is correct. So the first and the second are incorrect, so the option "Two of these are incorrect" is correct? Wait, no, the question is "Which of the following would be an incorrect way...". Wait, let's re - express the distance formula.

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Let's take \((x_1 = 2,y_1 = 3)\) and \((x_2=-4,y_2=-3)\).

- First option: \(\sqrt{(-4 - 2)^2+(3 - 3)^2}\). Here, \(x_2 - x_1=-4 - 2=-6\) (correct, since \(x_2=-4,x_1 = 2\)), but \(y_2 - y_1=-3 - 3=-6\), but in the formula, it is \(3 - 3 = 0\) (wrong).
- Second option: \(\sqrt{(2 - 4)^2+(3 - 3)^2}\). \(x_2\) is \(-4\), not \(4\), so \(x_1 - x_2=2-(-4)=6\), not \(2 - 4=-2\) (wrong), and \(y_2 - y_1\) is wrong as above.
- Third option: \(\sqrt{(2-(-4))^2+(3 - (-3))^2}\). \(x_1 - x_2=2-(-4)=6\), \(y_1 - y_2=3-(-3)=6\), which is correct (since \((x_1 - x_2)^2=(x_2 - x_1)^2\) and \((y_1 - y_2)^2=(y_2 - y_1)^2\)).

So the first option has a wrong \(y\) - component, the second option has wrong \(x\) and \(y\) - components. So two of them (first and second) are incorrect. Wait, but the fourth option says "Two of these are incorrect". But the question is asking which is an incorrect way. Wait, maybe I made a mistake. Wait, let's check the first option again. If we take \((x_1=-4,y_1 = - 3)\) and \((x_2 = 2,y_2=3)\), then \(x_2 - x_1=2-(-4)=6\), \(y_2 - y_1=3-(-3)=6\). But the first option is \(\sqrt{(-4 - 2)^2+(3 - 3)^2}=\sqrt{(-6)^2+0^2}\). The \(y\) - component here is \(3 - 3\), but \(y_2\) is \(3\) and \(y_1\) is \(-3\), so \(y_2 - y_1=3-(-3)=6\), not \(3 - 3\). So the first option is incorrect. The second option: \(\sqrt{(2 - 4)^2+(3 - 3)^2}\). The \(x\) - coordinates are \(2\) and \(-4\), not \(2\) and \(4\), so \(2 - 4\) is wrong, and \(y\) - component is wrong. The third option is correct. So the first and second are incorrect, so the option "Two of these are incorrect" is a correct statement? But the question is asking which is an incorrect way. Wait, maybe the options are:

Option 1: \(\sqrt{(-4 - 2)^2+(3 - 3)^2}\) - incorrect (wrong \(y\) - difference)

Option 2: \(\sqrt{(2 - 4)^2+(3 - 3)^2}\) - incorrect (wrong \(x\) and \(y\) - difference)

Option 3: \(\sqrt{(2-(-4))^2+(3 - (-3))^2}\) - correct

Option 4: Two of these are incorrect - correct (since option 1 and 2 are incorrect)

But the question is "Which of the following would be an incorrect way...", so we need to find the option that is an incorrect substitution. Wait, maybe I misread the options. Let's re - check the \(y\) - coordinate of the second point: it is \(-3\), not \(3\). So in option 1, the \(y\) - difference is \(3-(-3)\)? No, option 1 has \((3 - 3)\), which is wrong. Option 2 has \((2 - 4)\) (should be \(2-(-4)\)) and \((3 - 3)\) (should be \(3-(-3)\)). Option 3 is correct. So option 1 and option 2 are incorrect. So the option "Two of these are incorrect" is a correct statement, but the question is asking for an incorrect way. Wait, maybe the answer is that two of them are incorrect, so the option "Two of these are incorrect" is the answer? Wait, no, let's see:

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Let's substitute \((x_1 = 2,y_1 = 3)\) and \((x_2=-4,y_2=-3)\):

- Correct substitution 1: \(\sqrt{(-4 - 2)^2+(-3 - 3)^2}=\sqrt{(-6)^2+(-6)^2}\)
- Correct substitution 2: \(\sqrt{(2-(-4))^2+(3 - (-3))^2}=\sqrt{(6)^2+(6)^2}\) (which is option 3)

Now, option 1: \(\sqrt{(-4 - 2)^2+(3 - 3)^2}=\sqrt{(-6)^2+(0)^2}\) (incorrect, because \(y_2 - y_1=-3 - 3=-6\), not \(3 - 3 = 0\))

Option 2: \(\sqrt{(2 - 4)^2+(3 - 3)^2}=\sqrt{(-2)^2+(0)^2}\) (incorrect, because \(x_2=-4\), so \(x_1 - x_2=2-(-4)=6\), not \(2 - 4=-2\), and \(y_2 - y_1=-3 - 3=-6\), not \(3 - 3 = 0\))

Option 3: Correct

So option 1 and option 2 are incorrect. So the option "Two of these are incorrect" is a correct statement, but the question is asking which is an incorrect way. Wait, maybe the answer is that two of them are incorrect, so the option "Two of these are incorrect" is the answer. But let's check the options again.

Wait, the options are:

1. \(\sqrt{(-4 - 2)^2+(3 - 3)^2}\)

2. \(\sqrt{(2 - 4)^2+(3 - 3)^2}\)

3. \(\sqrt{(2-(-4))^2+(3 - (-3))^2}\)

4. Two of these are incorrect.

We saw that option 1 and option 2 are incorrect, so option 4 is correct. But the question is "Which of the following would be an incorrect way...", so if we consider that option 4 is a statement about the other options, and we need to find the incorrect substitution, then options 1 and 2 are incorrect, so the answer is that two of them are incorrect, i.e., option "Two of these are incorrect" is the answer. But maybe I made a mistake. Wait, let's check the \(y\) - coordinate again. The second point is \((-4,-3)\), so \(y_2=-3\). So in option 1, the \(y\) - difference is \(3 - 3\) (if we take \(y_1 = 3\) and \(y_2 = 3\), which is wrong, because \(y_2=-3\)). In option 2, \(x_2\) is taken as \(4\) instead of \(-4\), and \(y_2\) is taken as \(3\) instead of \(-3\). So both option 1 and option 2 are incorrect, so the option "Two of these are incorrect" is correct.

Answer:

Two of these are incorrect.