8th Grade
How do you use the Pythagorean Theorem to find the distance between two points in a coordinate plane?
{
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"manuscript": {
"title": {
"text": "How do you use the Pythagorean Theorem to find the distance between two points in a coordinate plane?",
"audio": "How do you use the Pythagorean Theorem to find the distance between two points in a coordinate plane?"
},
"description": {
"text": "To find the distance between two points in a coordinate plane, follow these four steps:\nOne, make a right triangle using the two points.\nTwo, find the lengths of the horizontal and vertical sides, a and b.\nThree, use the Pythagorean Theorem: a squared plus b squared equals c squared.\nFour, solve for c, the distance.",
"audio": "To find the distance between two points in a coordinate plane, follow these four steps. One, make a right triangle using the two points. Two, find the lengths of the horizontal and vertical sides, a and b. Three, use the Pythagorean Theorem: a squared plus b squared equals c squared. Four, solve for c, the distance."
},
"scenes": [
{
"text": "Let\u2019s try it with an example.\nTwo points, A and B, are placed here in the coordinate plane. Point A has the coordinates (1, 1), and Point B has the coordinates (4, 5).\nNow you\u2019re asked to find the distance between Point A and Point B. Let\u2019s label the distance c.",
"latex": ""
},
{
"text": "You’ll go through the same four steps as before, starting with step one: make a right triangle using the two points.",
"latex": ""
},
{
"text": "The Pythagorean Theorem applies to right triangles, so start by sketching a right triangle with the segment from A to B as the diagonal. From each point, draw horizontal and vertical lines until they meet at a right angle\u2014this forms the two legs of the triangle.",
"latex": ""
},
{
"text": "The horizontal side goes from x = 1 to x = 4, so the length of this side is 3. Let\u2019s label this side a. So, the length of a equals 3.",
"latex": ""
},
{
"text": "The vertical side goes from y = 1 to y = 5, so the length of this side is 4. Let\u2019s call this side b. So, the length of b equals 4.",
"latex": ""
},
{
"text": "Now for step three: use the Pythagorean Theorem, which states that a\u00b2 + b\u00b2 = c\u00b2\u2014in other words, the sum of the squares of the two shorter sides equals the square of the longest side, called the hypotenuse.",
"latex": "a^2 + b^2 = c^2"
},
{
"text": "Now plug in the values: 3 squared plus 4 squared equals c squared. That\u2019s 9 plus 16, which equals 25. To find c, you take the square root of 25. The square root of 25 is 5. That means the distance from one point to the other is 5 units.",
"latex": "3^2 + 4^2 = c^2 \\Rightarrow 9 + 16 = 25 \\Rightarrow c = \\sqrt{25} = 5"
},
{
"text": "Let\u2019s try one more example. This time, Point C is at (1, 4), and Point D is at (6, 8). You’ll go through the same four steps.",
"latex": ""
},
{
"text": "Step one: draw a right triangle connecting the two points.",
"latex": ""
},
{
"text": "Step two: find the side lengths. From x = 1 to x = 6 is a horizontal distance of 5, so a = 5. From y = 4 to y = 8 is a vertical distance of 4, so b = 4.",
"latex": "a = 5,\\ b = 4"
},
{
"text": "Step three: apply the Pythagorean Theorem. 5 squared plus 4 squared equals c squared. That\u2019s 25 plus 16, which equals 41.",
"latex": "5^2 + 4^2 = c^2 \\Rightarrow 25 + 16 = 41"
},
{
"text": "Step four: take the square root of 41. The square root of 41 is about 6.4. So, the distance between Point C and Point D is approximately 6.4 units.",
"latex": "c = \\sqrt{41} \\approx 6.4"
}
],
"outro": {
"text": "To find the distance between two points in a coordinate plane, follow these four steps:\nOne, make a right triangle using the two points.\nTwo, find the lengths of the horizontal and vertical sides, a and b.\nThree, use the Pythagorean Theorem: a squared plus b squared equals c squared.\nFour, solve for c, the distance.",
"audio": "To find the distance between two points in a coordinate plane, follow these four steps. One, make a right triangle using the two points. Two, find the lengths of the horizontal and vertical sides, a and b. Three, use the Pythagorean Theorem: a squared plus b squared equals c squared. Four, solve for c, the distance."
}
}
}
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