Question - Solving Right Triangle GHK
Solution:
The image shows two right triangles: Triangle GHK (which is triangle number 4) and another triangle next to it (triangle number 5), for which the details are not visible.In triangle GHK, we have to solve for variables x, y, and z. Triangle GHK is a right triangle, with the right angle at H.Given:- Segment GH = 5 (which is the base of the triangle).- Segment HK = 15 (which is the hypotenuse of the triangle).- Segment GH is perpendicular to segment HK, forming the right angle at H.- Segment GJ = x (this segment is not visible in the provided image, but since J is not defined, this could be a typo and might refer to GH).- Segment HJ = z (this segment is not visible in the provided image, but since J is not defined, this could be a typo and might refer to HJ, the height).- Segment JK = y (which is the remaining side, opposite the right angle).To solve for x, y, and z, we will use the Pythagorean theorem for right triangles: a^2 + b^2 = c^2, where a and b are the legs of the triangle, and c is the hypotenuse.Let's assume z refers to the height of the triangle (segment HJ), x refers to the length of the hypotenuse (segment HK, already given as 15), and y refers to the length of the opposite side of the right angle, which is usually the hypotenuse but seems to be used differently in this context. The given length of the hypotenuse (HK) as 15 and the base (GH) as 5 doesn't require solving and thus, x = 15 from the given information. To solve for y and z, you would proceed as follows:Since GH is 5, HJ (height, which is z) completing the perpendicular side, can be calculated using the Pythagorean theorem with values 5 and 15:z^2 = 15^2 - 5^2z^2 = 225 - 25z^2 = 200z = √200z = √(100 * 2)z = 10√2The length of z is 10√2.If y refers to the base opposite the height in this triangle, and with segment GH being part of segment GJ and the whole segment GJ (base) sums to 20 since it's the difference between HK (15) and HG (5); hence, segment JH will be:y = 15 - 5 = 10So the three variables solved from this triangle are:x = 15 (the hypotenuse, given)y = 10 (the length of JK, assuming JK refers to the remainder of the base)z = 10√2 (the height of the triangle)
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