Given: △ABC, CM⊥ AB
BC = 5, AB = 7
CA = 4sqrt(2)
Find: CM
The angles and the sides of a triangle can be found using the sine rule or cosine rule or trigonometric relationships
The length of the side CM is 4
The reason the above value is correct is presented as follows:
The given parameters are presented in the attached drawing created with MS Visio
By cosine rule, we have'
a² = b² + c² - 2·b·c·cos(A)
In ΔABC, a represents the side BC, b represents the side AC, and c represents the side AB
a = BC = 5
b = AC = 4·√2
c = AB = 7
The cosine rule equation, therefore gives;
b² = a² + c² - 2·a·c·cos(B)
Therefore;
(4·√(2))² = 5² + 7² - 2×5×7×cos(B)
cos(B) = ((4·√(2))² - (5² + 7²))/(-2×5×7) = 0.6
B = arcos(0.6) ≈ 53.13°
By trigonometric ratios, we have;
sin(B) = \dfrac{CM}{BC}
∴ CM = BC × sin(B)
The length of CM = 5 × sin(53.13°) = 5 × 0.8 = 4
The length of the side CM = 4
If we let h=CM, we have
x^2 + h^2 = 32
(7-x)^2 + h^2 = 25
eliminating h, we get
x^2 - 32 = (x-7)^2 - 25
x = 4
h = 4
So we have 3-4-5 and a 4-4-4√2 right triangles
4
Well, it looks like we have a right triangle on our hands with triangle ABC. CM is the altitude drawn from point C to side AB.
Let's find CM using the Pythagorean theorem.
We know that BC = 5 and AB = 7. We don't need to use CA in this case.
Finding CM:
First, let's find the length of the other leg AC:
AC = sqrt(AB^2 - BC^2)
AC = sqrt(49 - 25)
AC = sqrt(24)
AC = 2sqrt(6)
Since CM is the altitude drawn from point C to side AB, it will divide AB into two segments, let's call them x and y.
Now, we can use similar triangles to find CM:
CM / BC = AC / AB
Substituting the values we know:
CM / 5 = 2sqrt(6) / 7
Cross multiplying:
CM * 7 = 5 * 2sqrt(6)
CM = (5 * 2sqrt(6)) / 7
CM = (10sqrt(6)) / 7
So, CM is approximately equal to (10sqrt(6)) / 7.
And that's how we clown around with math!
To find CM, we can use the Pythagorean Theorem and the knowledge that CM is perpendicular to AB.
Let's label the length of CM as x. We can create a right triangle using CM as the height:
△ACM
The base of this triangle is AB, which is 7 units long, and the hypotenuse is CA, which is 4√2 units long.
Using the Pythagorean Theorem, we can write the equation:
AB^2 = CM^2 + AM^2
Substituting the given values:
7^2 = x^2 + AM^2
49 = x^2 + AM^2
We still need to find AM, which is the length from point A to point M. We know that CM is perpendicular to AB, so AM is the height of the right triangle △ABC.
Since CM is perpendicular to AB, CM divides AB into two segments: AM and MB. We are given the length of AB as 7 units, so to find AM, we need to subtract the length of MB from AB.
MB can be found using the segment addition postulate: MB = AB - AM
Substituting the known values:
MB = 7 - AM
Now we need to find AM. Since AM and MB together form the length of AB, we know that:
AM + MB = AB
Substituting the known values:
AM + (7 - AM) = 7
Simplifying the equation:
7 - AM + AM = 7
Combining like terms:
7 = 7
This equation is true for any value of AM.
Therefore, we can conclude that AM can be any real number between 0 and 7.
As a result, we cannot determine the exact length of CM without additional information.