Question - Calculating Lengths in a Right Triangle
Solution:
The image shows a coordinate plane with two points plotted on it: (3, 7) and (8, 5). The question asks to graph a right triangle with those two points forming the hypotenuse and calculate the distance between the two points (the length of the hypotenuse), as well as the lengths of the other sides.To form a right triangle, we need to find a point that is perpendicular to the hypotenuse and intersects at one of the axes. Given the coordinates, we can easily see that the right angle will be at the point where x or y coordinate is the same as one of the given points.Let's create the right triangle by plotting a third point that has the same x-coordinate as one point and the same y-coordinate as the other. We can take the x-coordinate from (8, 5) and the y-coordinate from (3, 7) to get the new point, which will be (8, 7).Now, to calculate the distances:1. The horizontal side (base) of the triangle is the difference between the x-coordinates of (3, 7) and (8, 7), which is 8 - 3 = 5 units.2. The vertical side (height) of the triangle is the difference between the y-coordinates of (3, 7) and (8, 5), which is 7 - 5 = 2 units.To find the distance between the two original points (the length of the hypotenuse), we use the Pythagorean theorem:$$ c = \sqrt{a^2 + b^2} $$where c is the hypotenuse, a is one leg, and b is the other leg of the triangle.Plugging in our values:$$ c = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} $$$$ \sqrt{29} $$ is already in its simplest radical form, but if we want to find the decimal value, it is approximately 5.4.So the length of the hypotenuse is approximately 5.4 units (to the nearest tenth). The lengths of the other two sides are 5 units and 2 units, respectively.
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