Question

given: t(-8, 12), v(-12, 7) find: tv
given: s(-1, 2), t(-10, 11) find: st

Explanation:

First Sub - Question: Find \( TV \) given \( T(-8,12) \) and \( V(-12,7) \)

Step 1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For points \( T(-8,12) \) (so \( x_1=-8,y_1 = 12 \)) and \( V(-12,7) \) (so \( x_2=-12,y_2=7 \)).

Step 2: Substitute values into the formula

First, calculate \( x_2 - x_1=-12-(-8)=-12 + 8=-4 \) and \( y_2 - y_1=7 - 12=-5 \).
Then, \( (x_2 - x_1)^2=(-4)^2 = 16 \) and \( (y_2 - y_1)^2=(-5)^2=25 \).
Sum these two results: \( 16 + 25=41 \)? Wait, no, wait: Wait, \((-4)^2=16\), \((-5)^2 = 25\), \(16+25 = 41\)? Wait, no, wait, I made a mistake. Wait, \(x_2 - x_1=-12-(-8)=-4\), \(y_2 - y_1=7 - 12=-5\). Then \((x_2 - x_1)^2=(-4)^2 = 16\), \((y_2 - y_1)^2=(-5)^2 = 25\). Then the sum is \(16 + 25=41\)? Wait, but the options have \(4\sqrt{13}\), \(\sqrt{202}\) etc. Wait, no, wait, maybe I misread the points. Wait, \(T(-8,12)\), \(V(-12,7)\). Let's recalculate: \(x_1=-8\), \(x_2=-12\), so \(x_2 - x_1=-12-(-8)=-4\). \(y_1 = 12\), \(y_2=7\), so \(y_2 - y_1=7 - 12=-5\). Then \((x_2 - x_1)^2=(-4)^2=16\), \((y_2 - y_1)^2=(-5)^2 = 25\). Then the distance is \(\sqrt{16 + 25}=\sqrt{41}\)? But that's not in the options. Wait, maybe the points are different? Wait, maybe \(T(-8,12)\) and \(V(-12, - 7)\)? No, the given is \(V(-12,7)\). Wait, maybe I made a mistake. Wait, let's check the second sub - question first.

Second Sub - Question: Find \( ST \) given \( S(-1,2) \) and \( T(-10,11) \)

Step 1: Use the distance formula

The distance formula is \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For \( S(-1,2) \) (\(x_1=-1,y_1 = 2\)) and \( T(-10,11) \) (\(x_2=-10,y_2=11\)).

Step 2: Calculate the differences

\(x_2 - x_1=-10-(-1)=-10 + 1=-9\) and \(y_2 - y_1=11 - 2 = 9\).

Step 3: Square the differences and sum

\((x_2 - x_1)^2=(-9)^2=81\) and \((y_2 - y_1)^2=9^2 = 81\).
Sum: \(81+81 = 162\).

Step 4: Take the square root

\(\sqrt{162}=\sqrt{81\times2}=9\sqrt{2}\).

Now, going back to the first sub - question. Let's re - check the points \(T(-8,12)\) and \(V(-12,7)\). \(x_1=-8\), \(x_2=-12\), so \(x_2 - x_1=-4\). \(y_1 = 12\), \(y_2=7\), so \(y_2 - y_1=-5\). Then \((x_2 - x_1)^2=16\), \((y_2 - y_1)^2 = 25\). Wait, \(16 + 25=41\), but that's not in the options. Wait, maybe the points are \(T(-8,12)\) and \(V(-12,-7)\)? Then \(y_2 - y_1=-7 - 12=-19\), \((y_2 - y_1)^2 = 361\), \(x_2 - x_1=-4\), \((x_2 - x_1)^2=16\), sum is \(377\), not helpful. Wait, maybe \(T(-8,12)\) and \(V(4,7)\)? Then \(x_2 - x_1=4-(-8)=12\), \(y_2 - y_1=7 - 12=-5\), \((x_2 - x_1)^2 = 144\), \((y_2 - y_1)^2=25\), sum is \(169\), \(\sqrt{169}=13\), and \(4\sqrt{13}\approx14.42\), no. Wait, maybe \(T(-8,12)\) and \(V(-12, - 7)\) is wrong. Wait, the option has \(4\sqrt{13}\), let's see what \(4\sqrt{13}\) squared is \(16\times13 = 208\). So we need \((x_2 - x_1)^2+(y_2 - y_1)^2=208\). Let's assume \(x_2 - x_1=-4\) (as before), then \((y_2 - y_1)^2=208 - 16=192\), \(y_2 - y_1=\pm\sqrt{192}=\pm8\sqrt{3}\), not 5. Wait, maybe \(x_2 - x_1=-12-(-8)=-4\) (correct), \(y_2 - y_1=7 - 12=-5\) (correct). Wait, maybe the problem is written wrong? Or maybe I misread the points. Alternatively, maybe \(T(-8,12)\) and \(V(4, - 7)\)? No. Wait, let's check the first sub - question again.

Wait, maybe the first sub - question's points are \(T(-8,12)\) and \(V(4, - 7)\)? No, the given is \(V(-12,7)\). Alternatively, maybe the distance formula was applied incorrectly. Wait, distance formula is \(d=\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}\), which is the same as \(\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) because squaring eliminates the sign. So that's not the issue.

Wait, maybe the first sub - question's answer is \( \sqrt{(-12 + 8)^2+(7 - 12)^2}=\sqrt{(-4)^2+(-5)^2}=\sqrt{16 + 25}=\sqrt{41}\), but it's not in the options. But the second sub - question's answer is \(9\sqrt{2}\) as we calculated.

Answer:

For \( TV \): (If we assume a miscalculation or misprint, but based on given points, \(\sqrt{41}\), but if we consider the option \(4\sqrt{13}\approx14.42\), \(\sqrt{41}\approx6.4\), not matching. Maybe there is a mistake in the problem statement.)

For \( ST \): \(9\sqrt{2}\)