Question

question 9(multiple choice worth 1 points)
(04.01 mc)
△efg is located at e (0, 0), f (-7, 4), and g (0, 8). which statement correctly classifies △efg?
△efg is a scalene triangle.
△efg is an isosceles triangle.
△efg is an equilateral triangle.
△efg is a right triangle.

question 10(multiple choice worth 1 points)
(04.01 mc)
quadrilateral abcd has coordinates a (3, 1), b (4, 4), c (7, 5), d (6, 2). quadrilateral abcd is a
rectangle, because opposite sides are congruent and adjacent sides are perpendicular.
square, because all four sides are congruent and adjacent sides are perpendicular.
parallelogram, because opposite sides are congruent and adjacent sides are not perpendicular.
rhombus, because all four sides are congruent and adjacent sides are not perpendicular.

Explanation:

Question 9

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 $EF$ with $E(0,0)$ and $F(-7,4)$:
$EF=\sqrt{(-7 - 0)^2+(4 - 0)^2}=\sqrt{49 + 16}=\sqrt{65}$
For $EG$ with $E(0,0)$ and $G(0,8)$:
$EG=\sqrt{(0 - 0)^2+(8 - 0)^2}=8$
For $FG$ with $F(-7,4)$ and $G(0,8)$:
$FG=\sqrt{(0+7)^2+(8 - 4)^2}=\sqrt{49+16}=\sqrt{65}$
Since $EF = FG=\sqrt{65}$, it is an isosceles triangle.

Question 10

Step1: Calculate side - lengths using distance formula

For $AB$ with $A(3,1)$ and $B(4,4)$:
$AB=\sqrt{(4 - 3)^2+(4 - 1)^2}=\sqrt{1 + 9}=\sqrt{10}$
For $BC$ with $B(4,4)$ and $C(7,5)$:
$BC=\sqrt{(7 - 4)^2+(5 - 4)^2}=\sqrt{9 + 1}=\sqrt{10}$
For $CD$ with $C(7,5)$ and $D(6,2)$:
$CD=\sqrt{(6 - 7)^2+(2 - 5)^2}=\sqrt{1+9}=\sqrt{10}$
For $DA$ with $D(6,2)$ and $A(3,1)$:
$DA=\sqrt{(3 - 6)^2+(1 - 2)^2}=\sqrt{9 + 1}=\sqrt{10}$

Step2: Calculate slopes of adjacent sides

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$
Slope of $AB$, $m_{AB}=\frac{4 - 1}{4 - 3}=3$
Slope of $BC$, $m_{BC}=\frac{5 - 4}{7 - 4}=\frac{1}{3}$
$m_{AB}\times m_{BC}=3\times\frac{1}{3}=1
eq - 1$, so adjacent sides are not perpendicular. Since opposite sides are congruent and adjacent sides are not perpendicular, it is a parallelogram.

Answer:

Question 9: B. ΔEFG is an isosceles triangle.
Question 10: C. parallelogram, because opposite sides are congruent and adjacent sides are not perpendicular.