Question

1. open - ended name three points on the coordinate plane that when connected form a right triangle. then explain how to calculate the length of the hypotenuse.

Explanation:

Step1: Choose three points

Let the points be \(A(0,0)\), \(B(0,3)\), \(C(4,0)\). The right - angle is at point \(A\) since one side lies on the \(y\) - axis and the other on the \(x\) - axis.

Step2: 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 a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), we can also use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\).

Step3: Identify the legs

The length of the vertical leg \(AB\) with \(A(0,0)\) and \(B(0,3)\): Using the distance formula \(d_{AB}=\sqrt{(0 - 0)^2+(3 - 0)^2}=3\). The length of the horizontal leg \(AC\) with \(A(0,0)\) and \(C(4,0)\): \(d_{AC}=\sqrt{(4 - 0)^2+(0 - 0)^2}=4\).

Step4: Calculate the hypotenuse

Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 3\) and \(b = 4\). So \(c=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\).

Answer:

Three points can be \(A(0,0)\), \(B(0,3)\), \(C(4,0)\). To calculate the length of the hypotenuse, first find the lengths of the two legs using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), then use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\) where \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. The length of the hypotenuse for the points chosen above is \(5\).