this is pathegreom theorem
22. Points A, B, and C are corners of a triangular field where mABC is 90°, AB is 40 meters and BC is 45 meters.
a) Find the length of AC .
A
40 m
B 45 m C b) John walks along the edge of the field from point A to point C. If P is
the point on AC when John is nearest to point B, find the length of BP.
|AC|^2 = 40^2 + 45^2 = 1600+2025
so
|AC| = 60.2 meters
Now this is the same as the problem below this one
draw a line through A perpendicular to AC
Now with trig the problem is trivial, but I will assume you must stick to geometry
let AP = x
then
CP = (60.2 - x)
call the altitude we want h
then
h^2 + x^2 = 40^2 = 1600
h^2 +(60.2-x)^2 = 45^2 = 2025
so
(60.2-x)^2 -x^2 = 2025-1600
3625 -120.4x = 425
x = 26.57
well so
26.57^2 + h^2 = 1600
h = 29.9
Hey, give me the whole problem the first time. When I did the one below you did not say angle was right angle at B
To find the length of AC, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, we have a right triangle ABC, where angle B is 90 degrees. The side opposite the right angle is AC, the hypotenuse.
We are given that AB is 40 meters and BC is 45 meters. We can use these lengths to find the length of AC.
To find the length of AC using the Pythagorean theorem, we can use the formula:
AC^2 = AB^2 + BC^2
Substituting the given values:
AC^2 = 40^2 + 45^2
AC^2 = 1600 + 2025
AC^2 = 3625
Taking the square root of both sides:
AC = √3625
AC ≈ 60.17 meters (rounded to two decimal places)
Therefore, the length of AC is approximately 60.17 meters.
Now, to find the length of BP, we need to determine the point on AC where John is nearest to point B. Since John is walking from point A to point C, he will be nearest to point B when he is perpendicular to side AC.
To find the length of BP, we can use the fact that the altitude of a right triangle from the right angle to the hypotenuse divides the hypotenuse into two segments that are proportional to the corresponding sides.
Using the similarity of triangles ABC and ABP (where P is the point on AC nearest to point B), we can set up the following proportion:
BP/AB = BC/AC
Substituting the given values:
BP/40 = 45/60.17
Cross-multiplying:
BP * 60.17 = 40 * 45
BP * 60.17 = 1800
Dividing both sides by 60.17:
BP = 1800/60.17
BP ≈ 29.92 meters (rounded to two decimal places)
Therefore, the length of BP is approximately 29.92 meters.