Answer:
True statements:
- You make a 90° turn.
- The distance labeled d₃ is the longest individual distance.
Step-by-step explanation:
To determine if a 90° turn is made, we need to calculate the slopes of the sides of the triangle. If the product of the slopes of two sides is -1, then the sides are perpendicular, which means the angle between them is 90°.
To calculate the slopes of sides of the triangle, we can use the slope formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Slope formula}}\\\\m=\dfrac{y_2-y_1}{x_2-x_1}\\\\\textsf{where:}\\\phantom{w}\bullet\;\;m\; \textsf{is the slope.}\\\phantom{w}\bullet\;\;(x_1,y_1)\;\textsf{and}\;(x_2,y_2)\;\textsf{are two points on the line.}\end{array}}[/tex]
Substitute the endpoints of each side of the triangle into the slope formula:
[tex]\textsf{Slope of $d_1$}=\dfrac{40-60}{65-5}=\dfrac{-20}{60}=-\dfrac{1}{3}[/tex]
[tex]\textsf{Slope of $d_2$}=\dfrac{-5-40}{50-65}=\dfrac{-45}{-15}=3[/tex]
[tex]\textsf{Slope of $d_3$}=\dfrac{60-(-5)}{5-50}=\dfrac{65}{-45}=-\dfrac{13}{9}[/tex]
As the product of the slopes of d₁ and d₂ is -1, then these sides are perpendicular, which means the angle between them is 90°. Therefore, a 90° turn is made between leg d₁ and d₂ of the journey.
Now, calculate the lengths of each leg of the route of the journey by using the distance formula:
[tex]\boxed{\begin{array}{l}\underline{\sf Distance \;Formula}\\\\d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\;d\;\textsf{is the distance between two points.} \\\phantom{ww}\bullet\;\;\textsf{$(x_1,y_1)$ and $(x_2,y_2)$ are the two points.}\end{array}}[/tex]
Therefore:
[tex]d_1=\sqrt{(65-5)^2+(40-60)^2}\\\\d_1=\sqrt{(60)^2+(-20)^2}\\\\d_1=\sqrt{3600+400}\\\\d_1=\sqrt{4000}\\\\d_1\approx 63.25\; \rm yd[/tex]
[tex]d_2=\sqrt{(50-65)^2+(-5-40)^2}\\\\d_2=\sqrt{(-15)^2+(-45)^2}\\\\d_2=\sqrt{225+2025}\\\\d_2=\sqrt{2250}\\\\d_2\approx 47.43\; \rm yd[/tex]
[tex]d_3=\sqrt{(5-50)^2+(60-(-5))^2}\\\\d_3=\sqrt{(-45)^2+(65)^2}\\\\d_3=\sqrt{2025+4225}\\\\d_3=\sqrt{6250}\\\\d_3\approx 79.06\; \rm yd[/tex]
So, the shortest distance from the house to the store is d₃, which is approximately 47.43 yd. This is also the longest individual distance.
The total distance of the journey is:
[tex]\textsf{Total distance}=d_1+d_2+d_3\\\\\textsf{Total distance}=\sqrt{4000}+\sqrt{2250}+\sqrt{6250}\\\\\textsf{Total distance}=189.7366596...\\\\\textsf{Total distance}\approx 189.74\; \rm yd[/tex]
As there are 1,760 yards in one mile, the bike ride is NOT over a 1 mile in total.
So, in summary, the true statements are:
- You make a 90° turn.
- The distance labeled d₃ is the longest individual distance.
[tex]\dotfill[/tex]
Additional Notes
A "unit of the grid" refers to the distance represented by one increment or division on the coordinate grid. In other words, it's the measure of length along the x-axis or y-axis between adjacent lines.
From observation of the given graph, the "unit of the grid" represents 10 units since every 2 lines are labelled in increments of 20, which means each line is an increment of 10. Therefore, "Each unit of the grid represents 10 yards" is simply a statement clarifying the scale used on the grid and does not necessitate any additional calculations.
