Question

find the distance between the points t(13, 1.6) and v(5.4, 3.7). answer in simplest exact form.

Explanation:

Step1: Recall distance formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 13\), \(y_1 = 1.6\), \(x_2 = 5.4\), \(y_2 = 3.7\).

Step2: Calculate differences

Find \(x_2 - x_1=5.4 - 13=-7.6\) and \(y_2 - y_1 = 3.7 - 1.6 = 2.1\).

Step3: Square the differences

\((x_2 - x_1)^2=(-7.6)^2 = 57.76\) and \((y_2 - y_1)^2=(2.1)^2 = 4.41\).

Step4: Sum the squares

Add the two squared values: \(57.76+4.41 = 62.17\).

Step5: Take the square root

\(d=\sqrt{62.17}\). Wait, but let's check the calculation again. Wait, \(x_1 = 13\), \(x_2 = 5.4\), so \(x_2 - x_1=5.4 - 13=-7.6\), squared is \((-7.6)^2 = 7.6\times7.6\). Let's recalculate \(7.6\times7.6\): \(7\times7 = 49\), \(7\times0.6 = 4.2\), \(0.6\times7 = 4.2\), \(0.6\times0.6 = 0.36\), so \((7 + 0.6)^2=7^2+2\times7\times0.6 + 0.6^2=49 + 8.4+0.36 = 57.76\). \(y_2 - y_1=3.7 - 1.6 = 2.1\), squared is \(4.41\). Sum is \(57.76+4.41 = 62.17\). So the distance is \(\sqrt{62.17}\). But maybe we can write it as \(\sqrt{\frac{6217}{100}}=\frac{\sqrt{6217}}{10}\) since \(62.17=\frac{6217}{100}\). Let's check if 6217 is a prime number. Let's test divisibility: 6217 divided by 13: \(13\times478 = 6214\), remainder 3. Divided by 7: \(7\times888 = 6216\), remainder 1. Divided by 17: \(17\times365 = 6205\), remainder 12. So \(\sqrt{62.17}=\frac{\sqrt{6217}}{10}\) or approximately, but the exact form is \(\sqrt{62.17}\) or simplified radical form as \(\frac{\sqrt{6217}}{10}\). Wait, maybe I made a mistake in coordinates? Wait the points are \(T(13, 1.6)\) and \(V(5.4, 3.7)\). So two - dimensional points. So the distance formula is correct. So the exact distance is \(\sqrt{(5.4 - 13)^2+(3.7 - 1.6)^2}=\sqrt{(-7.6)^2+(2.1)^2}=\sqrt{57.76 + 4.41}=\sqrt{62.17}\). If we want to write it as a fraction under the square root, \(62.17=\frac{6217}{100}\), so \(\sqrt{\frac{6217}{100}}=\frac{\sqrt{6217}}{10}\) (since \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) for \(a\geq0,b > 0\)).

Answer:

\(\sqrt{62.17}\) (or \(\frac{\sqrt{6217}}{10}\))