BM and CN ARE ALTITUDES OF TRIANGLE ABC.IF AB=3.5cm,AC=3.2cm,AND BM=2.1cm ,calculate the area of triangle ABC and hence find CN
area of triangle = (1/2)base x height
Using AC as the base, we are given BM as the height, so
area = (1/2)(3.2)(2.1) = 3.36
but we could have used AB as the base and CN as the height
area = (1/2)(3.5)(CN) = 3.36
CN = 3.36(2)/3.5 = ....
1.92
Well, it seems like we have a bit of a triangle puzzle here. Let's see what we can do!
To find the area of triangle ABC, we can use the formula for the area of a triangle: Area = 1/2 * base * height.
Since BM is an altitude of triangle ABC, we can use it as the height. And since AB is the base of the triangle, we can use it as the base. So, the area of triangle ABC is:
Area = 1/2 * AB * BM
= 1/2 * 3.5 cm * 2.1 cm
= 7.35 cm^2
Now that we have the area of triangle ABC, we can find the length of CN.
To do this, we can use the formula for the area of a triangle in terms of its sides: Area = 1/2 * AB * CN.
Using the given values, we can solve for CN:
7.35 cm^2 = 1/2 * 3.5 cm * CN
CN = 2 * (7.35 cm^2) / (3.5 cm)
CN ≈ 4.2 cm
So, the area of triangle ABC is 7.35 cm^2 and CN is approximately 4.2 cm.
To find the area of triangle ABC, we can use the formula:
Area = (1/2) * Base * Height
In this case, base AB = 3.5 cm and height BM = 2.1 cm.
Area = (1/2) * 3.5 cm * 2.1 cm
Area = 3.675 cm²
Now, to find CN, we can use the fact that the area of a triangle can also be calculated using the following formula:
Area = (1/2) * AB * CN
Substituting the values we know, we can solve for CN:
3.675 cm² = (1/2) * 3.5 cm * CN
Multiplying both sides by 2 and dividing by 3.5 cm, we get:
CN = (2 * 3.675 cm²) / 3.5 cm
CN = 2.05 cm
Therefore, the area of triangle ABC is 3.675 cm² and CN is 2.05 cm.
To calculate the area of triangle ABC, we can use the formula:
Area = (1/2) * base * height
In this case, the base of the triangle is AB, and the height is BM.
So, substituting the given values, we have:
Area = (1/2) * AB * BM
= (1/2) * 3.5cm * 2.1cm
= 3.675 cm^2
Now, to find CN, we can use the fact that in a triangle, the three altitudes intersect at a point called the orthocenter. Let's assume that this point is O.
Since the altitude BM is given, we can draw a line from point O perpendicular to BC, intersecting it at point N. This means that ON is the height of triangle BOC.
We can use the similarity between triangles ABC and BOC to find CN.
The sides of the triangles ABC and BOC are in proportion:
AB / BC = BM / CO
Substituting the given values, we have:
3.5cm / BC = 2.1cm / CO
Cross-multiplying, we get:
3.5cm * CO = 2.1cm * BC
Dividing both sides by 2.1cm, we get:
CO = (2.1cm * BC) / 3.5cm
Since the area of triangle ABC is given as 3.675 cm^2, we can use this to find BC. We know that the area of a triangle can be calculated as:
Area = (1/2) * BC * CN
Substituting the known values:
3.675 cm^2 = (1/2) * BC * CN
Rearranging the equation:
BC * CN = (2 * 3.675 cm^2) / 1
= 7.35 cm^2
Substituting the value of CO we found earlier:
(2.1cm * BC) / 3.5cm * CN = 7.35 cm^2
Cross-multiplying:
(2.1cm * CN) = (7.35 cm^2 * 3.5cm) / BC
Dividing both sides by 2.1cm:
CN = [(7.35 cm^2 * 3.5cm) / BC] / 2.1cm
Substituting the value of BC from before:
CN = [(7.35 cm^2 * 3.5cm) / (2.1cm * 3.675 cm^2)] / 2.1cm
Simplifying:
CN = 2.62 cm
Therefore, the area of triangle ABC is 3.675 cm^2 and CN is 2.62 cm.