ABC is a triangle with sides AB = 41 cm, BC = 28 cm and CA = 15 cm. If D, E and F are the mid-points of AB, BC and CA respectively, then what is the area of the triangle DEF?
31.5 square cm
Finding the Area of the Triangle Formed by Midpoints (Triangle DEF)
The problem asks us to find the area of triangle DEF, where D, E, and F are the mid-points of the sides AB, BC, and CA respectively, of triangle ABC. We are given the lengths of the sides of triangle ABC: AB = 41 cm, BC = 28 cm, and CA = 15 cm.
Understanding the Relationship Between Triangle ABC and Triangle DEF
Triangle DEF is known as the medial triangle of triangle ABC. A key property of the medial triangle is that its area is exactly one-fourth the area of the original triangle.
Mathematically, this relationship can be expressed as:
\(\text{Area(DEF)} = \frac{1}{4} \times \text{Area(ABC)}\)
Therefore, to find the area of triangle DEF, we first need to calculate the area of triangle ABC.
Calculating the Area of Triangle ABC using Heron's Formula
Since we are given the lengths of all three sides of triangle ABC, we can use Heron's formula to calculate its area. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by:
\(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)
where \(s\) is the semi-perimeter of the triangle, calculated as \(s = \frac{a+b+c}{2}\).
For triangle ABC, the side lengths are:
- a = BC = 28 cm
- b = CA = 15 cm
- c = AB = 41 cm
Step 1: Calculate the semi-perimeter (s)
\(s = \frac{28 + 15 + 41}{2}\)
\(s = \frac{84}{2}\)
\(s = 42 \text{ cm}\)
Step 2: Calculate (s-a), (s-b), and (s-c)
- \(s - a = 42 - 28 = 14 \text{ cm}\)
- \(s - b = 42 - 15 = 27 \text{ cm}\)
- \(s - c = 42 - 41 = 1 \text{ cm}\)
Step 3: Apply Heron's formula to find Area(ABC)
\(\text{Area(ABC)} = \sqrt{s(s-a)(s-b)(s-c)}\)
\(\text{Area(ABC)} = \sqrt{42 \times 14 \times 27 \times 1}\)
Let's simplify the expression under the square root:
\(42 = 2 \times 3 \times 7\)
\(14 = 2 \times 7\)
\(27 = 3 \times 3 \times 3 = 3^3\)
\(1 = 1\)
\(42 \times 14 \times 27 \times 1 = (2 \times 3 \times 7) \times (2 \times 7) \times (3^3) \times 1\)
\(= 2^2 \times 3^4 \times 7^2\)
Now, take the square root:
\(\text{Area(ABC)} = \sqrt{2^2 \times 3^4 \times 7^2}\)
\(\text{Area(ABC)} = 2^{\frac{2}{2}} \times 3^{\frac{4}{2}} \times 7^{\frac{2}{2}}\)
\(\text{Area(ABC)} = 2^1 \times 3^2 \times 7^1\)
\(\text{Area(ABC)} = 2 \times 9 \times 7\)
\(\text{Area(ABC)} = 18 \times 7\)
\(\text{Area(ABC)} = 126 \text{ square cm}\)
Calculating the Area of Triangle DEF
Now that we have the area of triangle ABC, we can find the area of triangle DEF using the property that Area(DEF) is one-fourth of Area(ABC).
\(\text{Area(DEF)} = \frac{1}{4} \times \text{Area(ABC)}\)
\(\text{Area(DEF)} = \frac{1}{4} \times 126\)
\(\text{Area(DEF)} = \frac{126}{4}\)
\(\text{Area(DEF)} = \frac{63}{2}\)
\(\text{Area(DEF)} = 31.5 \text{ square cm}\)
Thus, the area of triangle DEF is 31.5 square cm.
Revision Table: Triangle Area Concepts
| Concept | Description | Formula/Property |
|---|---|---|
| Semi-perimeter (s) | Half the sum of the side lengths of a triangle (a, b, c) | \(s = \frac{a+b+c}{2}\) |
| Heron's Formula | Used to find the area of a triangle when all three side lengths are known | \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\) |
| Midpoints of Triangle Sides | Points dividing each side into two equal halves | |
| Medial Triangle | The triangle formed by joining the midpoints of the three sides of a triangle (like DEF in this case) | |
| Area of Medial Triangle | The area of the medial triangle is 1/4th the area of the original triangle | \(\text{Area(Medial)} = \frac{1}{4} \times \text{Area(Original)}\) |
Additional Information: Medial Triangle Properties
The medial triangle (DEF) has several interesting properties related to the original triangle (ABC) besides the area relationship:
- The sides of the medial triangle are parallel to the corresponding sides of the original triangle. For example, DE is parallel to AC, EF is parallel to AB, and FD is parallel to BC.
- Each side of the medial triangle is half the length of the corresponding parallel side of the original triangle. For example, \(DE = \frac{1}{2}AC\), \(EF = \frac{1}{2}AB\), and \(FD = \frac{1}{2}BC\).
- The medial triangle divides the original triangle into four smaller triangles (the medial triangle itself and three other triangles at the corners). All four of these smaller triangles have the same area. This visually explains why the medial triangle's area is one-fourth of the original triangle's area.
- The perimeter of the medial triangle is half the perimeter of the original triangle.
These properties are consequences of the Midsegment Theorem, which states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side.
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