Given: △abc, cm ⊥ ab , bc=5, ab=7, ca=4sqaureroot2 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;

∴ CM = BC × sin(B)

The length of CM = 5 × sin(53.13°) = 5 × 0.8 = 4

The length of the side CM = 4

Learn more about cosine rule here:

link

We assume M is on AB, so that CM is the altitude to AB.

The triangle altitude can be found different ways. We choose to find the angle opposite the irrational side length using the Law of Cosines:

... b² = a² + c² - 2ac·cos(B)

... (4√2)² = 5² + 7² - 2·5·7·cos(B)

... 32 = 74 -70·cos(B)

... cos(B) = (74 -32)/70 = 3/5

Then

... BM = BC·cos(B) = 5·(3/5) = 3

By the Pythagorean theorem, ...

... BC² = BM² + CM²

... 5² = 3² + CM²

... CM = √(25 -9) = 4

The length of CM is 4.

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A diagram helps.

In the first diagram, we marked angle A to be 45° after realizing that AMC is an isosceles right triangle. This let us use a triangle solver with whole numbers to check the answer (second attachment). The area being 14 tells us the altitude to the side of length 7 is (2·14)/7 = 4, as we computed above.

sup

step-by-step explanation: