Answer:
a) 1650 m
b) 1677.05 m
Step-by-step explanation:
Hi there!
<u>1) Determine what is required for the answers</u>
For part A, we're asked for solve for the horizontal distance in which the road will rise 300 m. In other words, we're solving for the distance from point A to point C, point C being the third vertex of the triangle.
For part B, we're asked to solve for the length of the road, or the length of AB.
<u>2) Prove similarity</u>
In the diagram, we can see that there are two similar triangles: Triangle AXY and ABC (please refer to the image attached).
How do we know they're similar?
- Angles AYX and ACB are corresponding and they both measure 90 degrees
- Both triangles share angle A
Therefore, the two triangles are similar because of AA~ (angle-angle similarity).
<u>3) Solve for part A</u>
Recall that we need to find the length of AC.
First, set up a proportion. XY corresponds to BC and AY corresponds to AC:
![\frac{XY}{BC}=\frac{AY}{AC}](https://tex.z-dn.net/?f=%5Cfrac%7BXY%7D%7BBC%7D%3D%5Cfrac%7BAY%7D%7BAC%7D)
Plug in known values
![\frac{2}{300}=\frac{11}{AC}](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B300%7D%3D%5Cfrac%7B11%7D%7BAC%7D)
Cross-multiply
![2AC=11*300\\2AC=3300\\AC=1650](https://tex.z-dn.net/?f=2AC%3D11%2A300%5C%5C2AC%3D3300%5C%5CAC%3D1650)
Therefore, the road will rise 300 m over a horizontal distance of 1650 m.
<u>4) Solve for part B</u>
To find the length of AB, we can use the Pythagorean theorem:
where c is the hypotenuse of a right triangle and a and b are the other sides
Plug in 300 and 1650 as the legs (we are solving for the longest side)
![300^2+1650^2=c^2\\300^2+1650^2=c^2\\2812500=c^2\\1677.05=c](https://tex.z-dn.net/?f=300%5E2%2B1650%5E2%3Dc%5E2%5C%5C300%5E2%2B1650%5E2%3Dc%5E2%5C%5C2812500%3Dc%5E2%5C%5C1677.05%3Dc)
Therefore, the length of the road is approximately 1677.05 m.
I hope this helps!