SOLUTION: In the triangle y/x = 3. Which one of the following must be true? a. 4x>z b. 4x < z c. 4x=z d. it cannot be determined from the information given The picture of the triangle

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Question 568183: In the triangle y/x = 3. Which one of the following must be true?
a. 4x>z
b. 4x < z
c. 4x=z
d. it cannot be determined from the information given
The picture of the triangle shows one side is X one side is Y and one side is Z.
There is no right angle.the xy angle is obtuse, the zy and xz angles are acute.
I wish I knew how to draw the triangle for you but hopefully my description will do it.
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website!
Here is the way to analyze the situation. Since Z is opposite of the obtuse (biggest) angle, Z must be the longest side of the triangle. (This should be obvious from the drawing you were given for the triangle.)
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That means that the SUM of lengths of the two sides labeled X and Y must be greater than Z. (I believe that this is a theorem or an axiom.) Why is that true? Think about this. You cannot form a triangle if the lengths X and Y when added together are less than or just equal to the length of Z. If you need help picturing this, make this drawing:
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A. Draw a 4 inch long line about in the center of a sheet of paper.
B. Take a drawing compass and set the distance between its point and lead tip at 2 inches.
C. Place the compass point at one end of the 4 inch line swing a circle (of 2 inch radius) around that end of the line.
D. Then reset drawing compass so that the distance between its point and lead tip is 1 inch.
E. This time set the compass point at the other end of the 4 inch line and swing a circle (of 1 inch radius) around this end of the 4 inch line.
Notice that this is the case where the lengths of the two short sides (2 inch and 1 inch) total less than the long side (4 inches). The circles do not intersect meaning that the proposed length of 2 inches and 1 inch cannot be joined to form a triangle. If the radii of the two circles added up to be bigger than the longest side (for example one circle of radius 3 and the other of radius 2) then the circles would intersect and a triangle could be formed by drawing lines from the ends of the 4 inch line to the point where the circles intersected. One line to the intersection would be 3 units long and the other line would be 2 inches long, making a triangle of sides 2, 3, and 4.
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Once you can visualize and are convinced that X + Y has to be greater than Z for a triangle to be formed, you can write the inequality for a triangle having Z as its long side:
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Then you are told in the problem that:
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Solve this given equation for Y in terms of X by multiplying both sides by X to get:
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Now go to the inequality that we wrote above and in place of Y substitute its equal 3X to make the inequality become:
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Combine the two terms on the left side and you have the requirement that in the given triangle:
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This is answer a. in the list of answers that you were given.
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I hope this helps you to understand the problem that you were given and gives you a little more insight into how triangles are formed.
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