- Use the distance formula to compute the side lengths of the triangle: \(AB = 6\), \(BC = \sqrt{34}\), and \(AC = \sqrt{34}\).
- Observe that \(BC = AC\), which defines the triangle as isosceles.
- Conclude that Scott is incorrect and the correct classification is an isosceles triangle with legs of length \(\sqrt{34}\).
- Select option \(\boxed{a}\).
Explanation
- Calculate the side lengths of \(\triangle ABC\).
To analyze the triangle \(\triangle ABC\) with vertices \(A(2, 3)\), \(B(8, 3)\), and \(C(5, 8)\), we first calculate the lengths of its three sides: \(AB\), \(BC\), and \(AC\). The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
We will use this formula to determine the side lengths.,number:1,title:
- Compare the side lengths to classify the triangle.
- Length of side \(AB\):
Using \(A(2, 3)\) and \(B(8, 3)\):
\[AB = \sqrt{(8 - 2)^2 + (3 - 3)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6\]
- Length of side \(BC\):
Using \(B(8, 3)\) and \(C(5, 8)\):
\[BC = \sqrt{(5 - 8)^2 + (8 - 3)^2} = \sqrt{(-3)^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}\]
- Length of side \(AC\):
Using \(A(2, 3)\) and \(C(5, 8)\):
\[AC = \sqrt{(5 - 2)^2 + (8 - 3)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}\]
The side lengths are \(AB = 6\), \(BC = \sqrt{34}\), and \(AC = \sqrt{34}\).,number:2,title:
- Conclusion.
Since two sides have equal lengths (\(BC = AC = \sqrt{34}\)) and the third side is different (\(AB = 6\)), the triangle \(\triangle ABC\) is an isosceles triangle. Scott's claim that the triangle is equilateral is incorrect. Among the given options, we find that the triangle is isosceles with leg lengths of \(\sqrt{34}\), which corresponds to option a.,number:3,title:
Examples
In architecture and structural design, understanding the properties of triangles is essential for stability. If you are building a roof truss, you might need to determine if it is isosceles to ensure it is symmetrical and distributes loads evenly. For instance, knowing the side lengths allows you to calculate the angles using trigonometry, which helps in cutting the beams to the precise fit required for a sturdy frame. This application of geometry ensures that structures are both safe and aesthetically balanced.