Question

problem 6
calculate the length of each side of this right triangle
shorter len
longer leg
hypotenuse

Explanation:

Step1: Find shorter leg length

The two points for the shorter leg (vertical) are \((-10, -1)\) and \((-10, -6)\). The distance formula for vertical line is \(|y_2 - y_1|\). So, \(|-6 - (-1)| = |-5| = 5\).

Step2: Find longer leg length

The two points for the longer leg (horizontal) are \((-10, -6)\) and \((-3, -6)\). The distance formula for horizontal line is \(|x_2 - x_1|\). So, \(|-3 - (-10)| = |7| = 7\).

Step3: Find hypotenuse length

Use Pythagorean theorem \(c = \sqrt{a^2 + b^2}\), where \(a = 5\), \(b = 7\). So, \(c = \sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74}\approx 8.6\) (but exact value is \(\sqrt{74}\)). Wait, wait, no: Wait, the vertical leg: from \(-1\) to \(-6\) is 5 units (since \(-1 - (-6)=5\) or \(|-6 - (-1)| = 5\)). Horizontal leg: from \(-10\) to \(-3\) is \(|-3 - (-10)| = 7\) units. Then hypotenuse: \(\sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74}\approx 8.6\). But wait, maybe I mixed shorter and longer? Wait 5 is shorter than 7, so shorter leg is 5, longer leg is 7, hypotenuse \(\sqrt{74}\). Wait, but let's recheck the coordinates. Wait the three points are \((-10, -1)\), \((-10, -6)\), \((-3, -6)\). So the right angle is at \((-10, -6)\). So the legs are vertical (from \((-10, -1)\) to \((-10, -6)\)): length is \(|-1 - (-6)| = 5\) (since y-coordinates: -1 and -6, difference is 5). Horizontal leg: from \((-10, -6)\) to \((-3, -6)\): x-coordinates: -10 and -3, difference is 7. Then hypotenuse: between \((-10, -1)\) and \((-3, -6)\): distance formula \(\sqrt{(-3 - (-10))^2 + (-6 - (-1))^2} = \sqrt{(7)^2 + (-5)^2} = \sqrt{49 + 25} = \sqrt{74}\approx 8.6\). So shorter leg is 5, longer leg is 7, hypotenuse \(\sqrt{74}\) (or approximately 8.6). Wait, but maybe the grid is such that each square is 1 unit. So yes, vertical leg: 5 units (from y=-1 to y=-6, 5 steps down), horizontal leg: 7 units (from x=-10 to x=-3, 7 steps right), hypotenuse: \(\sqrt{5^2 + 7^2} = \sqrt{74}\).

Answer:

Shorter leg: 5
Longer leg: 7
Hypotenuse: \(\sqrt{74}\) (or approximately 8.6)