Question

what type of triangle is formed by joining the points d(7, 3), e(8, 1), and f(4, -1)?
a. equilateral triangle
b. isosceles triangle
c. right triangle
d. acute scalene triangle
e. obtuse scalene triangle

Explanation:

Step1: Calculate length of DE

Use distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $D(7,3)$ and $E(8,1)$, $x_1 = 7,y_1 = 3,x_2 = 8,y_2 = 1$. Then $DE=\sqrt{(8 - 7)^2+(1 - 3)^2}=\sqrt{1+( - 2)^2}=\sqrt{1 + 4}=\sqrt{5}$.

Step2: Calculate length of EF

For points $E(8,1)$ and $F(4,-1)$, $x_1 = 8,y_1 = 1,x_2 = 4,y_2=-1$. Then $EF=\sqrt{(4 - 8)^2+( - 1 - 1)^2}=\sqrt{( - 4)^2+( - 2)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$.

Step3: Calculate length of DF

For points $D(7,3)$ and $F(4,-1)$, $x_1 = 7,y_1 = 3,x_2 = 4,y_2=-1$. Then $DF=\sqrt{(4 - 7)^2+( - 1 - 3)^2}=\sqrt{( - 3)^2+( - 4)^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step4: Check triangle - type

Since $DE^2+EF^2=(\sqrt{5})^2+(2\sqrt{5})^2 = 5 + 20=25=DF^2$, by Pythagorean theorem, it is a right - triangle.

Answer:

C. right triangle