Question

determine whether a triangle with the given vertices is a scalene, isosceles, or equilateral triangle. check all that apply.
(a) j(-8, 3), k(-1, 0), l(-1, 6)
(b) d(2, 0), e(-1, -2), f(2, 6)
(c) p(0, 2), q(1, 8), r(-5, 8)
scalene triangle
isosceles triangle
equilateral triangle

Explanation:

Step1: Recall Distance Formula

The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). We'll use this to find the side lengths of each triangle.

Part (a): Triangle \(JKL\) with \(J(-8,3)\), \(K(-1,0)\), \(L(-1,6)\)

Step2: Calculate \(JK\)

Substitute \(x_1=-8,y_1 = 3,x_2=-1,y_2 = 0\) into the distance formula:
\(JK=\sqrt{(-1 - (-8))^2+(0 - 3)^2}=\sqrt{(7)^2+(-3)^2}=\sqrt{49 + 9}=\sqrt{58}\)

Step3: Calculate \(KL\)

Substitute \(x_1=-1,y_1 = 0,x_2=-1,y_2 = 6\):
\(KL=\sqrt{(-1 - (-1))^2+(6 - 0)^2}=\sqrt{0 + 36}=6\)

Step4: Calculate \(JL\)

Substitute \(x_1=-8,y_1 = 3,x_2=-1,y_2 = 6\):
\(JL=\sqrt{(-1 - (-8))^2+(6 - 3)^2}=\sqrt{7^2+3^2}=\sqrt{49 + 9}=\sqrt{58}\)
Since \(JK = JL=\sqrt{58}\) and \(KL = 6\) (two sides equal), it's an isosceles triangle.

Part (b): Triangle \(DEF\) with \(D(2,0)\), \(E(-1,-2)\), \(F(2,6)\)

Step5: Calculate \(DE\)

Substitute \(x_1 = 2,y_1=0,x_2=-1,y_2=-2\):
\(DE=\sqrt{(-1 - 2)^2+(-2 - 0)^2}=\sqrt{(-3)^2+(-2)^2}=\sqrt{9 + 4}=\sqrt{13}\)

Step6: Calculate \(EF\)

Substitute \(x_1=-1,y_1=-2,x_2=2,y_2=6\):
\(EF=\sqrt{(2 - (-1))^2+(6 - (-2))^2}=\sqrt{3^2+8^2}=\sqrt{9 + 64}=\sqrt{73}\)

Step7: Calculate \(DF\)

Substitute \(x_1 = 2,y_1=0,x_2=2,y_2=6\):
\(DF=\sqrt{(2 - 2)^2+(6 - 0)^2}=\sqrt{0 + 36}=6\)
Since all sides \(\sqrt{13}\), \(\sqrt{73}\), \(6\) are different, it's a scalene triangle.

Part (c): Triangle \(PQR\) with \(P(0,2)\), \(Q(1,8)\), \(R(-5,8)\)

Step8: Calculate \(PQ\)

Substitute \(x_1 = 0,y_1=2,x_2=1,y_2=8\):
\(PQ=\sqrt{(1 - 0)^2+(8 - 2)^2}=\sqrt{1 + 36}=\sqrt{37}\)

Step9: Calculate \(QR\)

Substitute \(x_1=1,y_1=8,x_2=-5,y_2=8\):
\(QR=\sqrt{(-5 - 1)^2+(8 - 8)^2}=\sqrt{(-6)^2+0}=\sqrt{36}=6\)

Step10: Calculate \(PR\)

Substitute \(x_1 = 0,y_1=2,x_2=-5,y_2=8\):
\(PR=\sqrt{(-5 - 0)^2+(8 - 2)^2}=\sqrt{25 + 36}=\sqrt{61}\)
Since all sides \(\sqrt{37}\), \(6\), \(\sqrt{61}\) are different, it's a scalene triangle.

Answer:

(a)

• Scalene triangle: \(\square\) (uncheck)
• Isosceles triangle: \(\boldsymbol{\checkmark}\) (check)
• Equilateral triangle: \(\square\) (uncheck)

(b)

• Scalene triangle: \(\boldsymbol{\checkmark}\) (check)
• Isosceles triangle: \(\square\) (uncheck)
• Equilateral triangle: \(\square\) (uncheck)

(c)

• Scalene triangle: \(\boldsymbol{\checkmark}\) (check)
• Isosceles triangle: \(\square\) (uncheck)
• Equilateral triangle: \(\square\) (uncheck)