Verified Answer :
- Calculate the slope of \(\overline{KL}\) using \(\frac{y_2-y_1}{x_2-x_1}\) to get \(0\).
- Calculate the length of \(\overline{KL}\) using \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) to get \(7\).
- Calculate the slope of \(\overline{MK}\) to get \(1\).
- Calculate the length of \(\overline{MK}\) to get \(\sqrt{50}\).
- Identify that the triangle is scalene as all side lengths are distinct.
- Final values: \(\boxed{\text{Slope } KL=0, \text{Len } KL=7, \text{Slope } MK=1, \text{Len } MK=\sqrt{50}}\).
Explanation
- Analysis and Formulas
We are given the coordinates of three vertices of a triangle in a 2D plane: \(K(2, 0)\), \(L(9, 0)\), and \(M(7, 5)\). Our objective is to determine the slopes and lengths of the sides \(\overline{KL}\), \(\overline{LM}\), and \(\overline{MK}\).
The slope \(m\) of a line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
The length \(d\) of a line segment between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
We already know:
- Slope of \(\overline{LM} = \frac{5 - 0}{7 - 9} = \frac{5}{-2} = -2.5\)
- Length of \(\overline{LM} = \sqrt{(7 - 9)^2 + (5 - 0)^2} = \sqrt{(-2)^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}\)
- Calculations for KL and MK
For segment \(\overline{KL}\) with points \(K(2, 0)\) and \(L(9, 0)\):
- Slope \(m_{KL} = \frac{0 - 0}{9 - 2} = \frac{0}{7} = 0\).
- Length \(d_{KL} = \sqrt{(9 - 2)^2 + (0 - 0)^2} = \sqrt{7^2 + 0^2} = \sqrt{49} = 7\).
For segment \(\overline{MK}\) with points \(M(7, 5)\) and \(K(2, 0)\):
- Slope \(m_{MK} = \frac{5 - 0}{7 - 2} = \frac{5}{5} = 1\).
- Length \(d_{MK} = \sqrt{(7 - 2)^2 + (5 - 0)^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}\) (approximately \(7.07\)).
- Final Characterization
Based on our calculations, the properties of the triangle \(\triangle KLM\) are:
- Slope of \(\overline{KL} = 0\)
- Length of \(\overline{KL} = 7\)
- Slope of \(\overline{LM} = -2.5\)
- Length of \(\overline{LM} = \sqrt{29}\)
- Slope of \(\overline{MK} = 1\)
- Length of \(\overline{MK} = \sqrt{50}\)
The triangle has side lengths \(7\), \(\sqrt{29} \approx 5.39\), and \(\sqrt{50} \approx 7.07\). Since all side lengths are different, the triangle is a scalene triangle.
Examples
In urban planning, architects often plot landmarks on a grid to measure distances and calculate angles for infrastructure. By representing buildings as points \(K, L,\) and \(M\), they can determine the lengths of paths or the slope of terrain between them, ensuring accurate placement of pathways or utility lines. For instance, knowing the slope between two points helps in designing wheelchair-accessible ramps or determining drainage patterns for landscape architecture.