Question

question 5 of 5 select the correct answer. △def has vertices of d(1,1), e(4,5), and f(9, - 5). what is an accurate description of the triangle? right scalene acute scalene obtuse isosceles acute equilateral

Explanation:

Step1: Calculate side - lengths using distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side $DE$ with $D(1,1)$ and $E(4,5)$:
$DE=\sqrt{(4 - 1)^2+(5 - 1)^2}=\sqrt{3^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5$.
For side $EF$ with $E(4,5)$ and $F(9,-5)$:
$EF=\sqrt{(9 - 4)^2+(-5 - 5)^2}=\sqrt{5^2+(-10)^2}=\sqrt{25 + 100}=\sqrt{125}=5\sqrt{5}$.
For side $DF$ with $D(1,1)$ and $F(9,-5)$:
$DF=\sqrt{(9 - 1)^2+(-5 - 1)^2}=\sqrt{8^2+(-6)^2}=\sqrt{64+36}=\sqrt{100}=10$.
Since all side - lengths are different ($DE = 5$, $EF=5\sqrt{5}$, $DF = 10$), the triangle is scalene.

Step2: Check for right - angle using Pythagorean theorem

$DE^{2}+DF^{2}=25 + 100=125$, and $EF^{2}=(5\sqrt{5})^{2}=125$. So, by the Pythagorean theorem ($a^{2}+b^{2}=c^{2}$, where $c$ is the longest side), $\angle D = 90^{\circ}$.

Answer:

right scalene