Question

1. if \\( \overline { q n } \\) bisects \\( \angle p q r \\) and \\( n \\) is the midpoint of \\( \overline { p r } \\) classify each triangle by its angles and sides.

a. \\( \delta p q r \\) :
b. \\( \delta p r t \\) :
c. \\( \delta t q s \\) :
d. \\( \delta q n p \\) :
directions: find the measures of the sides of \\( \delta d e f \\), then classify it by its sides.

1. \\( d ( 8, - 6 ), e ( - 1, - 3 ), f ( - 2,5 ) \\)

Explanation:

Step1: Calculate the length of \(DE\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(D(8,-6)\) and \(E(-1,-3)\), \(x_1 = 8,y_1=-6,x_2=-1,y_2 = -3\).
\(DE=\sqrt{(-1 - 8)^2+(-3+6)^2}=\sqrt{(-9)^2+3^2}=\sqrt{81 + 9}=\sqrt{90}=3\sqrt{10}\)

Step2: Calculate the length of \(EF\)

For \(E(-1,-3)\) and \(F(-2,5)\), \(x_1=-1,y_1=-3,x_2=-2,y_2 = 5\).
\(EF=\sqrt{(-2 + 1)^2+(5 + 3)^2}=\sqrt{(-1)^2+8^2}=\sqrt{1+64}=\sqrt{65}\)

Step3: Calculate the length of \(DF\)

For \(D(8,-6)\) and \(F(-2,5)\), \(x_1 = 8,y_1=-6,x_2=-2,y_2 = 5\).
\(DF=\sqrt{(-2 - 8)^2+(5 + 6)^2}=\sqrt{(-10)^2+11^2}=\sqrt{100+121}=\sqrt{221}\)

Answer:

Since \(DE = 3\sqrt{10}\), \(EF=\sqrt{65}\), \(DF=\sqrt{221}\) and \(DE
eq EF
eq DF\), \(\triangle DEF\) is a scalene triangle.