Let ABC be a triangle. If D(2, 5) and E(5, 9) are the mid-points of the sides AB and AC respectively, then what is the length of the side BC?
10
Solving the Triangle Midpoint Problem
The problem asks us to find the length of side BC in a triangle ABC, given the coordinates of the midpoints of sides AB and AC. The midpoints are D(2, 5) on AB and E(5, 9) on AC.
This problem can be solved using the Midpoint Theorem (also known as the Midsegment Theorem). The theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side.
In our case, DE is the segment connecting the midpoints D of AB and E of AC. Therefore, according to the Midpoint Theorem, DE is parallel to BC, and the length of DE is half the length of BC. This means:
\(\text{Length of BC} = 2 \times \text{Length of DE}\)
Calculating the Length of the Midsegment DE
To find the length of DE, we use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\), which is given by:
\(\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
The coordinates of D are \((x_1, y_1) = (2, 5)\) and the coordinates of E are \((x_2, y_2) = (5, 9)\).
Let's plug these values into the distance formula:
\(\text{Length of DE} = \sqrt{(5 - 2)^2 + (9 - 5)^2}\)
\(\text{Length of DE} = \sqrt{(3)^2 + (4)^2}\)
\(\text{Length of DE} = \sqrt{9 + 16}\)
\(\text{Length of DE} = \sqrt{25}\)
\(\text{Length of DE} = 5\)
So, the length of the midsegment DE is 5 units.
Finding the Length of Side BC using Midpoint Theorem
Now that we have the length of DE, we can use the relationship from the Midpoint Theorem:
\(\text{Length of BC} = 2 \times \text{Length of DE}\)
Substitute the length of DE we found:
\(\text{Length of BC} = 2 \times 5\)
\(\text{Length of BC} = 10\)
Therefore, the length of the side BC is 10 units.
Summary of Steps
- Identify that D and E are midpoints and relate DE to BC using the Midpoint Theorem.
- Calculate the distance between points D and E using the distance formula.
- Multiply the length of DE by 2 to find the length of BC.
Revision Table: Key Formulas
| Concept | Formula | Application |
|---|---|---|
| Distance Formula (between \((x_1, y_1)\) and \((x_2, y_2)\)) | \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) | Used to find the length of segment DE. |
| Midpoint Theorem | Midsegment connecting two sides' midpoints is half the length of the third side. | Used to relate the length of DE to the length of BC (BC = 2 * DE). |
Additional Information on Midpoint Theorem
The Midpoint Theorem is a powerful tool in coordinate geometry and Euclidean geometry. It establishes a clear relationship between the midsegment of a triangle and the third side.
- The midsegment is always parallel to the third side. This property is useful in proving lines are parallel.
- The length of the midsegment is exactly half the length of the third side. This property allows us to find the length of one segment if the other is known.
- A triangle has three midsegments, connecting the midpoints of each pair of sides. These three midsegments divide the original triangle into four smaller triangles that are congruent to each other and similar to the original triangle.
Understanding the Midpoint Theorem is crucial for solving problems involving midpoints of triangle sides and lengths or parallelism of segments.
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