Question

use the right triangle abc to find the length of the line segment between point a and point b. now, find the exact length of the hypotenuse, c. that’s the length of the line segment between point a and point b. use square root notation to write the exact value of c. you can solve below or solve on paper and then enter your answer on zearn. the coordinate of b is (-1, 5), c is (7, 5), a is (7, -3). the formula is $a^2 + b^2 = c^2$, then $8^2 + 8^2 = c^2$, find c.

Explanation:

Step1: Calculate \(8^2 + 8^2\)

First, we know that \(8^2 = 64\), so \(8^2 + 8^2 = 64 + 64 = 128\). So the equation becomes \(128 = c^2\).

Step2: Solve for \(c\)

To find \(c\), we take the square root of both sides. Since \(c\) represents a length, we take the positive square root. So \(c = \sqrt{128}\). We can simplify \(\sqrt{128}\) by factoring 128: \(128 = 64\times2\), and \(\sqrt{64\times2}=\sqrt{64}\times\sqrt{2}=8\sqrt{2}\). But we can also leave it as \(\sqrt{128}\) or simplify it further. Wait, actually, from the equation \(8^2 + 8^2 = c^2\), \(64 + 64 = 128\), so \(c = \sqrt{128}=\sqrt{64\times2}=8\sqrt{2}\), but also, let's check the coordinates. Wait, point B is (-1,5), point C is (7,5), so the length BC is \(7 - (-1)=8\). Point C is (7,5), point A is (7,-3), so the length AC is \(5 - (-3)=8\). So triangle ABC is a right triangle with legs of length 8 and 8. Then by Pythagoras, \(c^2 = 8^2 + 8^2 = 64 + 64 = 128\), so \(c = \sqrt{128}=8\sqrt{2}\), but also, \(\sqrt{128}\) can be written as \(\sqrt{64\times2}=8\sqrt{2}\), or we can note that \(128 = 8\times16\)? Wait, no, 64 is 8 squared. Wait, 8 squared is 64, so 64 + 64 is 128. So the square root of 128 is \(8\sqrt{2}\), but also, \(\sqrt{128}=\sqrt{64\times2}=8\sqrt{2}\approx11.31\), but the exact value is \(8\sqrt{2}\) or \(\sqrt{128}\). Wait, but let's check the calculation again. Wait, 8 squared is 64, so 64 + 64 is 128, so \(c = \sqrt{128}\), which simplifies to \(8\sqrt{2}\) because 128 = 642, and sqrt(642)=sqrt(64)sqrt(2)=8sqrt(2). But also, \(\sqrt{128}\) can be written as \(8\sqrt{2}\), but maybe the problem just wants the simplified radical form or the exact value. Wait, let's compute \(8^2 + 8^2 = 128\), so \(c = \sqrt{128}\), which is equal to \(8\sqrt{2}\), but also, 128 is 816? No, 168 is 128, but 642 is 128. So the exact length is \(\sqrt{128}\) or \(8\sqrt{2}\). Wait, but let's check the steps again. Step 1: Compute \(8^2 + 8^2\): 64 + 64 = 128. Step 2: Take square root: \(c = \sqrt{128}=8\sqrt{2}\). But also, \(\sqrt{128}\) can be simplified as \(8\sqrt{2}\), but maybe the problem accepts \(\sqrt{128}\) or \(8\sqrt{2}\). Wait, let's see, 8 squared is 64, so 64 + 64 is 128, so \(c = \sqrt{128}\), which is equal to \(8\sqrt{2}\) (since 128 = 642, and sqrt(642)=8sqrt(2)). So the exact length is \(8\sqrt{2}\) or \(\sqrt{128}\). But let's confirm with the coordinates. The horizontal distance between B and C is 8 (from x=-1 to x=7), vertical distance between C and A is 8 (from y=5 to y=-3), so right triangle with legs 8 and 8, so hypotenuse is \(8\sqrt{2}\). So the exact value of \(c\) is \(8\sqrt{2}\) (or \(\sqrt{128}\), but \(8\sqrt{2}\) is simplified).

Answer:

\(8\sqrt{2}\) (or \(\sqrt{128}\), but \(8\sqrt{2}\) is the simplified radical form)