Calculating the Area of Triangle BDE

The question asks for the area of triangle BDE, given that the area of triangle ABC is 44 cm², D is the midpoint of BC, and E is the midpoint of AB.

Understanding the Properties of Midpoints

When D is the midpoint of side BC, it means that BD = DC, so BD is half the length of BC.

When E is the midpoint of side AB, it means that AE = EB, so BE is half the length of AB.

Relating Triangle BDE to Triangle ABC

Consider triangle BDE and triangle ABC. Both triangles share angle B.

• Side BE of $\Delta$BDE is half of side AB of $\Delta$ABC ($\text{BE} = \frac{1}{2}\text{AB}$).
• Side BD of $\Delta$BDE is half of side BC of $\Delta$ABC ($\text{BD} = \frac{1}{2}\text{BC}$).
• Angle B is common to both triangles.

Using the Area Formula for Triangles

The area of a triangle can be calculated using the formula: $\text{Area} = \frac{1}{2} \times \text{side}_1 \times \text{side}_2 \times \sin(\text{included angle})$.

For $\Delta$ABC:

$\text{Area}(\Delta\text{ABC}) = \frac{1}{2} \times \text{AB} \times \text{BC} \times \sin(\text{B})$

For $\Delta$BDE:

$\text{Area}(\Delta\text{BDE}) = \frac{1}{2} \times \text{BE} \times \text{BD} \times \sin(\text{B})$

Substituting the Midpoint Relationships

Substitute the relationships $\text{BE} = \frac{1}{2}\text{AB}$ and $\text{BD} = \frac{1}{2}\text{BC}$ into the area formula for $\Delta$BDE:

$\text{Area}(\Delta\text{BDE}) = \frac{1}{2} \times \left(\frac{1}{2}\text{AB}\right) \times \left(\frac{1}{2}\text{BC}\right) \times \sin(\text{B})$

$\text{Area}(\Delta\text{BDE}) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \text{AB} \times \text{BC} \times \sin(\text{B})$

$\text{Area}(\Delta\text{BDE}) = \frac{1}{4} \times \left(\frac{1}{2} \times \text{AB} \times \text{BC} \times \sin(\text{B})\right)$

Notice that the term in the parenthesis is the area of $\Delta$ABC.

So, $\text{Area}(\Delta\text{BDE}) = \frac{1}{4} \times \text{Area}(\Delta\text{ABC})$.

Calculating the Area of Triangle BDE

Given that $\text{Area}(\Delta\text{ABC}) = 44 \text{ cm}^2$, we can calculate the area of $\Delta$BDE:

$\text{Area}(\Delta\text{BDE}) = \frac{1}{4} \times 44 \text{ cm}^2$

$\text{Area}(\Delta\text{BDE}) = 11 \text{ cm}^2$

Conclusion

The area of triangle BDE is 11 cm².

Revision Table: Triangle Area and Midpoints

Additional Information: Midpoint Theorem Context

The Midpoint Theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half its length.

In $\Delta$ABC, since E is the midpoint of AB and D is the midpoint of BC, the line segment ED is parallel to AC and $\text{ED} = \frac{1}{2}\text{AC}$.

If we were to also find the midpoint of AC, say F, and connect D, E, and F, the triangle ABC would be divided into four smaller triangles: $\Delta$BDE, $\Delta$AED, $\Delta$DFC, and $\Delta$EFD.

A key property is that these four triangles ($\Delta$BDE, $\Delta$AED, $\Delta$DFC, $\Delta$EFD) are congruent to each other. Therefore, they all have the same area.

Since there are four such congruent triangles that make up the original triangle ABC, the area of each smaller triangle is one-fourth the area of $\Delta$ABC.

Our specific problem focuses on $\Delta$BDE, which is indeed one of these four triangles (if we imagine the line ED being drawn, even if the midpoint of AC isn't explicitly mentioned). The relationship $\text{Area}(\Delta\text{BDE}) = \frac{1}{4} \times \text{Area}(\Delta\text{ABC})$ holds true, as proven by both the area formula method and the property arising from the Midpoint Theorem when all three midpoints are considered.