Area of Triangle ABC and the Triangle formed by Joining Midpoints of the Sides

To find the area of triangle ABC, we need to use the information given about the triangle formed by joining the midpoints of its sides. Let's break down the problem step by step:

Step 1: Understanding the Midsegment Theorem

The triangle formed by joining the midpoints of the sides of triangle ABC is known as the medial triangle. According to the Midsegment Theorem, the midsegment of a triangle is parallel to the third side of the triangle, and its length is half the length of the third side.

Step 2: Identifying the Midpoints

To find the area of triangle ABC, we first need to identify the midpoints of its sides. Let's label the midpoints as D, E, and F. So, line segment DE is the midsegment for side AB, EF is the midsegment for side BC, and FD is the midsegment for side AC.

Step 3: Using the Midsegment Theorem

Since the midsegment is half the length of the third side, we can conclude that:

- DE = (1/2) AB
- EF = (1/2) BC
- FD = (1/2) AC

Step 4: Finding the Area of the Medial Triangle

We are given that the area of the medial triangle (formed by joining the midpoints) is 2 square units. Let's label the vertices of the medial triangle as P, Q, and R, corresponding to the midpoints D, E, and F, respectively.

To find the area of the medial triangle, we can use the formula for the area of a triangle, which is given by:

Area = (1/2) base * height

In this case, the base of the medial triangle is PR, and the height is the perpendicular distance from Q to PR. Let's say this distance is h.

Therefore, the area of the medial triangle is:

2 = (1/2) PR * h

Simplifying the equation, we get:

PR * h = 4

Step 5: Relationship between the Medial Triangle and Triangle ABC

It is important to note that the medial triangle is similar to triangle ABC. The sides of the medial triangle are parallel to the sides of triangle ABC, and their lengths are half of the corresponding sides of triangle ABC.

Step 6: Finding the Area of Triangle ABC

Given that the area of the medial triangle is 2 square units, we can find the area of triangle ABC by multiplying the area of the medial triangle by 4 since the lengths of the sides are doubled.

Therefore, the area of triangle ABC is:

Area of ABC = 4 * 2 = 8 square units.

Conclusion

In conclusion, the area of triangle ABC is 8 square units. This was found by using the Midsegment Theorem to establish the relationship between the medial triangle and triangle ABC. The given information about the area of the medial triangle allowed us to calculate the area of triangle ABC.

8 sq unit