Question

station
classify each triangle as scalene, isosceles, or equilateral. determine whether the triangle is a right triangle.

1. j(-7, -7), k(-9, 1), l(-1, -1)
2. j(-3, 2), k(2, 6), l(8, -1)

Explanation:

Step1: Calculate the lengths of the sides using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \(JK\) with \(J(-7,-7)\) and \(K(-9,1)\):

$$
LATEXBLOCK0
$$

For \(JL\) with \(J(-7,-7)\) and \(L(-1,-1)\):

$$
LATEXBLOCK1
$$

For \(KL\) with \(K(-9,1)\) and \(L(-1,-1)\):

$$
LATEXBLOCK2
$$

Step2: Classify the triangle

Since \(JK = KL=2\sqrt{17}\), the triangle is isosceles.

Step3: Check for right - triangle using Pythagoras theorem \(a^{2}+b^{2}=c^{2}\)

Let \(a = 2\sqrt{17}\), \(b = 2\sqrt{17}\), \(c = 6\sqrt{2}\)
\(a^{2}+b^{2}=(2\sqrt{17})^{2}+(2\sqrt{17})^{2}=68 + 68=136\)
\(c^{2}=(6\sqrt{2})^{2}=72\)
Since \(a^{2}+b^{2}
eq c^{2}\), it is not a right - triangle.

Step1: Calculate the lengths of the sides using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \(JK\) with \(J(-3,2)\) and \(K(2,6)\):

$$
LATEXBLOCK0
$$

For \(JL\) with \(J(-3,2)\) and \(L(8,-1)\):

$$
LATEXBLOCK1
$$

For \(KL\) with \(K(2,6)\) and \(L(8,-1)\):

$$
LATEXBLOCK2
$$

Step2: Classify the triangle

Since \(JK
eq JL
eq KL\), the triangle is scalene.

Step3: Check for right - triangle using Pythagoras theorem \(a^{2}+b^{2}=c^{2}\)

Let \(a=\sqrt{41}\), \(b = \sqrt{85}\), \(c=\sqrt{130}\)
\(a^{2}+b^{2}=41+85 = 126\)
\(c^{2}=130\)
Since \(a^{2}+b^{2}
eq c^{2}\), it is not a right - triangle.

Answer:

The triangle with vertices \(J(-7,-7)\), \(K(-9,1)\), \(L(-1,-1)\) is an isosceles triangle and not a right - triangle.

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