Problem: A triangular lot has two sides of length 100m and 48m as indicated in the figure below. The length of the perpendicular from a corner of the lot to the 48m side is 96m. A fence is to be erected perpendicular to the 48m side so that the area of the lot is equally divided. How far from A along segment AB should this perpendicular fence be constructed? Give your answer in simplest radical form AND rounded to the nearest tenth of a meter.
Image (figure that is given): gyazo(dot)com(forward slash)95d78114652ee0c58a6a2214bcbb4dc3
Help on this would be VERY much appreciated. I'm quite confused on the wording itself, all I've found out so far is that the area of the triangle is 2304m and that segment AD is 28m using pythag. thm. From this I also know that segment DB is 20m and that segment CB must be square root of 9616. This is where I get stuck and don't know how to advance ;l
The new perpendicualar must be parallel to CD
Let MN be that perpendicular with M on AC and N on AC
I agree that AD = 28, so we can set up a ratio:
MN/AM = 96/28 = 24/7
MN = 24AM/7
Area of triangle AMN = (1/2)(2304) = 1152
= (1/2)(AM)(MN)
(1/2)(AM)(24AM/7) = 1152
AM^2 = 672
am=sqrt(672)
= appr 25.9 m
check:
AM=25.9
MN = 24(25.9)/7 = 88.8
Area of triangle AMN
=(1/2)(25.9)(88.8)
= 1149.96
which is close enough to 1152, allowing for round-off
To solve this problem, we can use the concept of similar triangles and the area of a triangle. Here's the step-by-step explanation:
1. Start by finding the area of the large triangle ABC. Since the length of one of the sides (AB) and the length of the perpendicular (AD) are known, you can use the formula for the area of a triangle: A = (1/2) * base * height. So, A = (1/2) * 48m * 96m = 2304m^2.
2. Since we want to divide the area of the triangle equally, we need to find the area of each smaller triangle. Let's call the point where the fence meets the side AB as point X.
3. Notice that the smaller triangles ABX and ACB are similar, meaning their corresponding angles are equal. This is because the fence is perpendicular to the side AB.
4. We can use the property of similar triangles to find the length AX. The ratio of the sides of similar triangles is equal. So, (AX / AB) = (AC / CB).
5. We know that AX + XB = AB, and we can substitute the values in the ratio from step 4, giving us (AX / AB) = (AX / (AX + XB)) = ((AC / CB)).
6. Substituting the known values, we have AX / (AX + XB) = (48m / √9616m^2). Simplifying further, (AX / (AX + XB)) = 1 / (√9616m^2 / 48m).
7. Now, we can solve for AX using this equation. Let's cross-multiply to get AX * (√9616m^2 / 48m) = AX + XB.
8. Rearrange the equation to isolate AX by subtracting AX from both sides: AX * (√9616m^2 / 48m) - AX = XB.
9. Factor out AX: AX * (√9616m^2 / 48m - 1) = XB.
10. Simplify the expression inside the parentheses: (√9616m^2 - 48m) / 48m.
11. Substitute the known values: AX * [(√9616m^2 - 48m) / 48m] = XB.
12. Substitute the known values of AB and AD: AX * [(√9616m^2 - 48m) / 48m] = 48m - 28m.
13. Simplify the equation to find AX: AX * [(√9616m^2 - 48m) / 48m] = 20m.
14. Divide both sides by the expression [(√9616m^2 - 48m) / 48m] to isolate AX: AX = (20m * 48m) / (√9616m^2 - 48m).
15. Simplify further: AX = 960m / (√9616m^2 - 48m).
16. Now, you can use the formula to calculate the value of AX. Calculate √9616m^2 - 48m and simplify the expression. Then plug it into the formula to find the numerical value of AX.
17. Round the result to the nearest tenth of a meter, as requested.
Following these steps, you should be able to find the distance along segment AB where the perpendicular fence should be constructed.