Triangle JKM has j=28cm, k=17cm, and m=13cm.
a. find the measure of the smallest angle
b. find the area of the triangle
To find the measure of the smallest angle in triangle JKM, we can use the Law of Cosines. The Law of Cosines states that in any triangle, the square of one side equals the sum of the squares of the other two sides minus twice the product of their lengths and the cosine of the included angle.
In this case, we want to find the smallest angle, so let's use the sides j = 28cm, k = 17cm, and m = 13cm.
First, we'll find the largest angle, which we'll call angle JKM. Using the Law of Cosines, we can calculate it as follows:
cos(JKM) = (j^2 + k^2 - m^2) / (2 * j * k)
cos(JKM) = (28^2 + 17^2 - 13^2) / (2 * 28 * 17)
cos(JKM) = (784 + 289 - 169) / (2 * 28 * 17)
cos(JKM) = 904 / 952
cos(JKM) ≈ 0.948529
Now, we can use the inverse cosine function (arccos) to find the measure of angle JKM:
JKM = arccos(0.948529)
JKM ≈ 19.17 degrees
Since we want the smallest angle, which would be one of the other two angles, we can subtract angle JKM from 180 degrees to find the measure of the smallest angle:
Smallest Angle = 180 - JKM
Smallest Angle ≈ 160.83 degrees
So, the measure of the smallest angle is approximately 160.83 degrees.
To find the area of the triangle, we can use Heron's formula. Heron's formula states that the area of a triangle with sides a, b, and c is given by:
Area = sqrt(s * (s - a) * (s - b) * (s - c))
where s is the semiperimeter of the triangle, defined as:
s = (a + b + c) / 2
In this case, the sides of the triangle are j = 28cm, k = 17cm, and m = 13cm. Let's calculate the semiperimeter first:
s = (28 + 17 + 13) / 2
s = 58 / 2
s = 29
Now we can calculate the area using Heron's formula:
Area = sqrt(29 * (29 - 28) * (29 - 17) * (29 - 13))
Area = sqrt(29 * 1 * 12 * 16)
Area = sqrt(696)
Area ≈ 26.37 cm²
Therefore, the area of the triangle is approximately 26.37 cm².
a) use law of cosines
b) use hero's formula