Triangle geometry find a side length

Wildcat! :smile:In summary, the problem involves finding the length of CB in triangle ABC, with given information about the median from C meeting AB at D, the midpoint of CD at M, and the midpoint of CP at P. A possible solution includes using the area method by dividing the triangle into smaller triangles and adding their areas, or using the theorem about mid-segments of a triangle to find a relationship between P and Q.
  • #1
Wildcat
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Homework Statement


In triangle ABC, the median from C meets AB at D. Through M, the midpoint of CD, line AM is drawn meeting CB at P. If CP=4, find CB.


Homework Equations





The Attempt at a Solution


I constructed this drawing on GSP and found CB to be 12. I'm trying to show similarity between some triangles in the drawing but can't find any. I would like to know how to solve this without GSP. any ideas??
 
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  • #2
Hi Wildcat! :smile:

Hint: areas. :wink:​
 
  • #3
Ok, I don't see where I can calculate any areas with the information I have unless I'm missing something. Will I need to construct another segment?
 
  • #4
Hi Wildcat! :smile:

(just got up :zzz:)
Wildcat said:
Will I need to construct another segment?

Yes.

Divide the triangle into triangles, call two of the unequal areas "p" and "q", and add them all up. :smile:​
 
  • #5
Wildcat said:

Homework Statement


In triangle ABC, the median from C meets AB at D. Through M, the midpoint of CD, line AM is drawn meeting CB at P. If CP=4, find CB.

Homework Equations


The Attempt at a Solution


I constructed this drawing on GSP and found CB to be 12. I'm trying to show similarity between some triangles in the drawing but can't find any. I would like to know how to solve this without GSP. any ideas??

Hi Wildcat,

Apart from the area method, there's still another way to tackle this problem. It's to use mid-segment of a triangle (it's the line segment that connects the two midpoints of any 2 sides of a triangle).

There are 2 theorems about mid-segment you should remember is:
Given [itex]\Delta ABC[/itex]
  • If M, and N are respectively the midpoints of AB, and AC then [itex]MN = \frac{1}{2}BC[/itex], and [itex]MN // BC[/itex].
    This theorem means that the mid-segment of a triangle is parallel to the opposite side, and is half of it.​
  • If a line passes through the midpoint of one side, and is parallel to the second side, then it also passes through the midpoint of the other side.

-------------------------------

So back to your problem,

Let d be a line that passes through D, and parallel to AM, it intersects BC at Q. Now, look at the 2 theorems above, what conclusion can you draw about P, and Q?

Hint: Look closely at the 2 triangles [itex]\Delta ABP[/itex], and [itex]\Delta CDQ[/itex]

Cheers,
 

FAQ: Triangle geometry find a side length

What is the Pythagorean Theorem and how does it relate to finding a missing side length in a triangle?

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be used to find a missing side length in a right triangle by rearranging the equation to solve for the missing side.

How do you use trigonometry to find a missing side length in a triangle?

Trigonometry involves using the ratios of the sides of a right triangle to find missing side lengths. In order to use trigonometry, you must know at least one angle and one side length of the triangle. The three basic trigonometric ratios are sine, cosine, and tangent, which can be used to find the missing side length depending on which ratio is needed for the given information.

Can you use the Law of Cosines to find a missing side length in any type of triangle?

The Law of Cosines can be used to find a missing side length in any triangle, regardless of its angles. It states that the square of one side length is equal to the sum of the squares of the other two sides, minus twice the product of those sides and the cosine of the included angle. This equation can be rearranged to solve for the missing side length.

How do you use the Law of Sines to find a missing side length in a triangle?

The Law of Sines can be used to find a missing side length in a triangle if you know the length of at least two sides and the measure of one angle. The law states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of a triangle. By setting up and solving a proportion using this ratio, you can find the missing side length.

Is there a way to find a missing side length in a triangle if you don't know any angles?

If you do not know any angles in a triangle, you can still use the distance formula to find a missing side length. This formula involves finding the distance between two points on a coordinate plane, which can be done using the Pythagorean Theorem. Once you have the distance, you can use it as a side length in a triangle and solve for the missing side using the Pythagorean Theorem again.

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