For this scalene triangle: which of the following is true?

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In summary, the problem attached as TheProblem.jpg has an answer of D. The equations attached in Metric relationships summary.pdf may be helpful. The line YW is not perpendicular to XZ, so the metric summary table does not fully apply. The question involves understanding the effect of bisecting an angle. One approach is to consider what conditions would make each statement true. Another approach is to carefully draw and measure scalene triangles. The answer D can be interpreted as the ratio of corresponding sides in similar triangles.
  • #1
s3a
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Homework Statement


The problem is attached as TheProblem.jpg and the answer is D.

Homework Equations


Perhaps the equations attached in Metric relationships summary.pdf.

The Attempt at a Solution


My logic is as follows:
For A, XW does not necessarily equal WZ.

For B, it seems that they are trying to trick the student into thinking of h^2 = d*e (from the Metric relationships summary.pdf file).

For C, the ratio does not work out because the angle facing each respective smaller/larger leg is different.

For D, I can't figure it out but it seems to relate to the Metric relationships summary.pdf file I attached.

Any input would be greatly appreciated!
Thanks in advance!
 

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  • #2
The metric summary table does not quite apply here because the line YW is not perpendicular to XZ.

What it shows you is that the angle at Y has been bisected - so that ∠XYW = ∠WYZ.
The question wants to know if you understand what this does to the relationships.

One way of thinking this through is to figure wat it would take for each condition to be true: what does it mean?

For instance, A is true if the two sub-triangles are similar (since they share a side, this means the overall triangle XYZ must be isosceles.)

Of course, another approach is to physically (and carefully) draw a few (large) scalene triangles, bisect one angle, and then measure the corresponding sides :)
 
  • #3
Thanks!

Thanks to you telling me to focus on the bisected angle, I basically converted the answer D into words to make sense of it:

(small leg of small triangle)/(hypotenuse of small triangle) = (small leg of large triangle)/(hypotenuse of large triangle).

The "hypotenuse of small triangle" part is technically not the hypotenuse of the small triangle but rather a length that is equivalently large as it.
 
  • #4
Great, well done.
Putting it in words is a pain - personally I relabel these things with lower case for side lengths and upper case for the corresponding angles.
 

1. What is a scalene triangle?

A scalene triangle is a type of triangle in which all three sides have different lengths. This means that no two sides are equal in length.

2. What are the properties of a scalene triangle?

A scalene triangle has no equal sides and no equal angles. It can have three acute angles, three obtuse angles, or one obtuse angle and two acute angles.

3. Which of the following is true for a scalene triangle?

A. It has three equal sides.
B. It has three equal angles.
C. It has two equal sides and one equal angle.
D. None of the above.

The correct answer is D. None of the above. A scalene triangle has no equal sides or angles.

4. Can a scalene triangle be a right triangle?

Yes, a scalene triangle can be a right triangle. This means that one of its angles measures 90 degrees.

5. How do you find the area of a scalene triangle?

To find the area of a scalene triangle, you can use the formula A = (1/2)bh, where b is the base of the triangle and h is its height. You can also use Heron's formula, which is more complex but can be used to find the area of any type of triangle.

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