Midsegment Theorem Math Example 4

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Example 4

hard
In △ABC\triangle ABC, the three midsegments are drawn, dividing the triangle into four smaller triangles. If the area of △ABC\triangle ABC is 120120 cm², what is the area of each smaller triangle? Justify using the Midsegment Theorem.

Solution

  1. 1
    Step 1: By the Midsegment Theorem, each midsegment is half the length of the side it's parallel to. The three midsegments create four smaller triangles that are all congruent to each other.
  2. 2
    Step 2: Each small triangle is similar to △ABC\triangle ABC with a linear scale factor of 12\frac{1}{2}. The area scale factor is (12)2=14\left(\frac{1}{2}\right)^2 = \frac{1}{4}.
  3. 3
    Step 3: Total area =120= 120 cm². Four congruent triangles share this area equally: each has area 1204=30\frac{120}{4} = 30 cm².

Answer

Each smaller triangle has area 3030 cm².
The three midsegments of a triangle divide it into four congruent triangles, each similar to the original with ratio 1:2. Since area scales as the square of the linear ratio, each small triangle has area (12)2=14(\frac{1}{2})^2 = \frac{1}{4} of the original. Four quarters account for the entire area.

About Midsegment Theorem

A segment connecting the midpoints of two sides of a triangle is parallel to the third side and exactly half its length.

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More Midsegment Theorem Examples

Example 1 easy

In [formula], [formula] is the midpoint of [formula] and [formula] is the midpoint of [formula]. If

Example 2 medium

In [formula], [formula] is the midpoint of [formula] and [formula] is the midpoint of [formula]. If

Example 3 easy

The midsegment of a triangle has length 14. What is the length of the side parallel to the midsegmen

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