Question 697144: Find w
In the right triangle, the opposite side is w , the adajcent is w+7 and the hypotenuse is w+9.
Can you please help me I don't understand? Thanks so much in advance.
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! This problem gives the measurements of the three sides of the right triangle in trigonometric terms. We are used to hearing or seeing that the right triangle has two legs and one hypotenuse (or the longest side). The major difference between just calling the legs "legs" and calling them "adjacent" and "opposite" is this: Adjacent and opposite legs are used when we deal with specific angles. These specific angles usually are given in the diagram or information. These are "relative" names. They are based on the position of the specific angle we are dealing with. This angle is either of acute angles in the triangle (angles of than the right angle) or both, but you will almost never have to do problems where that angle is the right angle. This is because we run into some complications with terminology if we used the right angle. Nonetheless, the angle (usually given as the Greek symbol theta in advanced work) that is used determines what legs are called adjacent and opposite. Adjacent basically means "next to" and opposite basically means "away from" or "on the other side." The adjacent side is the leg that is right next to the angle; the opposite side is the leg away from the angle. The hypotenuse is fixed. That is, no matter if we deal with calling them legs or this terminology, the hypotenuse is always the longest side. It is always opposite the right angle. It will always be next to the specific angles we deal with (assuming those angles are not the right angle). The adjacent and opposite angles change based on what angle we use.
Since we are dealing with sides (I'm assuming we are dealing with side measurement), we really do not need to worry about which leg is the adjacent one and which is the opposite one. The Pythagorean theorem doesn't need to know which is which, all it needs to know is that the sum of the squares of the two legs equals the square of the hypotenuse:
a^2 + b^2 = c^2.
And since we know our two legs are w and w + 7 and the hypotenuse is w + 9, we can solve for w:
w^2 + (w+7)^2 = (w+9)^2
w^2 + w^2 + 14w + 49 = w^2 + 18w + 81
2w^2 + 14w + 49 = w^2 + 18w + 81
w^2 + 14w + 49 = 18w + 81
w^2 - 4w + 49 = 81
w^2 - 4w - 32 = 0
(w-8)(w+4) = 0
We need the positive answer because we are dealing with measurement. So, w = 8. Now we just plug this into what we said the sides equaled:
Opposite = x = 8
Adjacent = x + 7 = 15
Hypotenuse = x + 9 = 17
Now if we are dealing with angles, then it is important to know which leg is which. However, by the information you've given me, I cannot determine where the angle is because you've given me no angle.
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