Question

practice

1. the grid represents a map of joses neighborhood. it shows the locations of his house as well as the houses of four friends.

a. draw a triangle between the houses of jose, ed, and brad. determine whether this triangle is a scalene, isosceles, or equilateral triangle. explain your reasoning.
b. determine whether the triangle is a right triangle. explain your reasoning. if it is not a right triangle, determine whether it is acute or obtuse.
c. jose, miles, and brad are meeting for band rehearsal. miles claims that the distance from joses house to his house is the same as the distance from joses house to brads house. is his claim correct? explain your answer. what kind of triangle is formed if you connect their houses?
d. a new boy, james, moved into the neighborhood at the location (-3, -5). plot and label jamess house on the grid. then, determine whether the triangle formed by connecting his house, joses house, and malcolms house is a right triangle.

Explanation:

Step1: Assume coordinates

Let's assume Jose's house is at point $J(x_1,y_1)$, Ed's house is at point $E(x_2,y_2)$ and Brad's house is at point $B(x_3,y_3)$. From the grid, if we assume Jose's house $J(- 4,3)$, Ed's house $E(-4,-1)$ and Brad's house $B(5,1)$.

Step2: Calculate side - lengths using distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$

For side $JE$:
$d_{JE}=\sqrt{(-4+4)^2+( - 1 - 3)^2}=\sqrt{0+( - 4)^2}=4$
For side $JB$:
$d_{JB}=\sqrt{(5 + 4)^2+(1 - 3)^2}=\sqrt{81 + 4}=\sqrt{85}$
For side $EB$:
$d_{EB}=\sqrt{(5 + 4)^2+(1 + 1)^2}=\sqrt{81+4}=\sqrt{85}$
Since $d_{JB}=d_{EB}
eq d_{JE}$, the triangle is isosceles.

Step3: Check for right - triangle using Pythagorean theorem $a^{2}+b^{2}=c^{2}$

Let $a = 4$, $b=\sqrt{85}$, $c=\sqrt{85}$.
$a^{2}+b^{2}=4^{2}+\sqrt{85}^{2}=16 + 85=101$
$c^{2}=\sqrt{85}^{2}=85$
Since $a^{2}+b^{2}
eq c^{2}$, it is not a right - triangle.
To check if it is acute or obtuse, we use the law of cosines $c^{2}=a^{2}+b^{2}-2ab\cos C$. If $c^{2}

Step4: For part c

Let Jose's house $J(-4,3)$, Miles' house $M(5,5)$ and Brad's house $B(5,1)$
Distance from Jose to Miles: $d_{JM}=\sqrt{(5 + 4)^2+(5 - 3)^2}=\sqrt{81 + 4}=\sqrt{85}$
Distance from Jose to Brad: $d_{JB}=\sqrt{(5 + 4)^2+(1 - 3)^2}=\sqrt{81 + 4}=\sqrt{85}$
Miles' claim is correct. The triangle formed is isosceles.

Step5: For part d

Plot James' house at $(-3,-5)$. Let Jose's house $J(-4,3)$ and Malcolm's house $M(0,-4)$
Distance $d_{JM}=\sqrt{(-3 + 4)^2+( - 5 - 3)^2}=\sqrt{1+64}=\sqrt{65}$
Distance $d_{JM'}=\sqrt{(0 + 3)^2+( - 4 + 5)^2}=\sqrt{9 + 1}=\sqrt{10}$
Distance $d_{M'M}=\sqrt{(0 + 4)^2+( - 4 - 3)^2}=\sqrt{16 + 49}=\sqrt{65}$
Using Pythagorean theorem: $(\sqrt{10})^{2}+(\sqrt{65})^{2}=10 + 65=75
eq(\sqrt{65})^{2}$, so it is not a right - triangle.

Answer:

a. The triangle is isosceles.
b. It is not a right - triangle. It is an acute triangle.
c. Miles' claim is correct. The triangle formed is isosceles.
d. It is not a right - triangle.