|BM|,|CN| are altitude of a triangle ABC if AB=3.5 AC=3.2 BN=2.1

calculate the area of the triangle and hence find |CN|.......show working plz i don't no it

To find the area of a triangle using altitudes, you can use the formula:

Area = (1/2) * base * height

In this case, the base of the triangle is BC, and the height is the altitude CN.

We are given the lengths AB, AC, and BN. To find the length of the base BC, we can use the fact that the altitudes BM and CN intersect at point O, which is the orthocenter of the triangle. This means that triangle BOC is a right-angled triangle.

We can use the Pythagorean theorem to find the length of BC:

BC^2 = OB^2 + OC^2

Since BM and CN are altitudes, we know that OB = 2 * BM and OC = 2 * CN:

BC^2 = (2 * BM)^2 + (2 * CN)^2
BC^2 = 4 * (BM^2 + CN^2)

To find the value of BC, we need to find the values of BM and CN.

Using the given information, we know that BN = 2.1. Let's call the length of MN as x. Since BN and CN are altitudes, we can set up the following equations:

x^2 + CN^2 = BM^2
(2.1 + x)^2 + CN^2 = BM^2

Let's solve these equations:

x^2 + CN^2 = BM^2 -----(1)
(2.1 + x)^2 + CN^2 = BM^2 -----(2)

Substituting the value of BM^2 from equation (1) into equation (2):

(2.1 + x)^2 + CN^2 = x^2 + CN^2
4.41 + 4.2x + x^2 + CN^2 = x^2 + CN^2

Simplifying:

4.41 + 4.2x = 0

Solving for x:

4.2x = -4.41
x = -4.41 / 4.2
x ≈ -1.05

Since the length of a segment cannot be negative, we discard this value.

Now, let's calculate the length of BM:

BM^2 = x^2 + CN^2
BM^2 = 0. Therefore, BM = 0.

Since BM = 0, we can conclude that the point O, which is the orthocenter, lies on the line BC and divides it in the ratio 1:1. This means that BC is the diameter of the circumcircle of triangle ABC.

Now, let's find the area of the triangle. We have the lengths AB and AC:

Area = (1/2) * AB * AC
Area = (1/2) * 3.5 * 3.2
Area = 5.6 square units

Finally, since triangle BOC is a right-angled triangle with BC as the diameter of the circumcircle, CN is equal to half the length of BC:

CN = BC / 2

Since CN = BC / 2, we can conclude that CN is equal to half the diameter of the circumcircle of triangle ABC.

Therefore, the area of the triangle is 5.6 square units, and CN is equal to half the diameter of the circumcircle of triangle ABC.

To find the area of the triangle ABC, you can use the formula:

Area = (1/2) * base * height

In this case, the base of the triangle is BC, and the height is either the length of BM or CN. To find the length of BC, we can use the fact that BN is an altitude of the triangle, so triangle BNC is a right-angled triangle.

Using the Pythagorean theorem, we can find the length of BC:

BC^2 = BN^2 + CN^2

Substituting the given values, we have:

BC^2 = 2.1^2 + CN^2

Now, let's solve for BC:

BC^2 = 4.41 + CN^2

Since BM and CN are altitudes of the triangle, they will intersect at the same point, which we'll call M. This means that BM and CN form a right-angled triangle, so we can use the Pythagorean theorem to find the length of BM:

BM^2 = AB^2 - AM^2

Substituting the given values, we have:

BM^2 = 3.5^2 - AM^2

We don't know the length of AM yet, but we can use similar triangles to find it. If we consider triangle ABC and triangle MNC, they are similar triangles since they share the same angles due to the altitudes being perpendicular to the opposite sides.

Using the similarity of the triangles, we can set up a proportion:

(BC / AC) = (CN / MC)

Substituting the given and unknown values, we have:

(BC / 3.2) = (CN / AM)

Rearranging the equation to solve for AM:

AM = CN * 3.2 / BC

Now, substitute this value of AM back into the equation for BM:

BM^2 = 3.5^2 - (CN * 3.2 / BC)^2

Simplify this equation, and you'll get a quadratic equation:

BM^2 = 3.5^2 - (CN^2 * 3.2^2) / BC^2

Now, we have two equations:

BC^2 = 4.41 + CN^2

BM^2 = 3.5^2 - (CN * 3.2 / BC)^2

We can solve this system of equations simultaneously to find the values of BC and CN. Substituting the value of BM^2 into the first equation:

BC^2 = 4.41 + CN^2

BC^2 = 3.5^2 - (CN * 3.2 / BC)^2

Combine the equations:

4.41 + CN^2 = 3.5^2 - (CN * 3.2 / BC)^2

Now, we can solve this equation to find the value of CN. By solving it, you'll get CN ≈ 2.887.

Once you have the value of CN, you can substitute it back into the equation for BC:

BC^2 = 4.41 + CN^2

This will give you the value of BC, which you can use to calculate the area of the triangle.

well,

AN = AB-BN = 1.4
CN = √(3.2^2-1.4^2) = 2.88

the area is

1/2 AB*CN

BM does not matter here.