The sides AB and AC of triangle ABC have lengths of 4 inches and 7 inches, respectively. The median AM is 3.5 inches. What is the EXACT length of BC. Show your work.
extend AM its own length of 3.5 to point D.
Join BD and CD
You now have a quadrilateral where the diagonals AD and BC bisect each other.
So ABDC must be a parallelogram .
In triangle ADC you could now use the cosine law to find angle DAC
Once you have that you can use the cosine law again to find MC in triangle AMC
double MC to find BC
Triangle ABC, A at the top, B & C at the bottom, B left & C right.
The length of a triangle median is defined by
(Ma)^2 = (b^2 + c^2)/2 - a^2/4. (Ma is the median from vertex A to side "a".)
(Mb)^2 = (a^2 + c^2)/2 - b^2/4
(Mc)^2 = (a^2 + b^2)/2 - c^2/4
Letting BC = a, AB = c = 4, AC = b = 7 and AH = 3.5
Then, (3.5^2) = (7^2 + 4^2)/2 - a^2/4
Multiplying through by 4 yields 4(3.5)^2 = 2(49 + 16) - a^2 or
49 = 130 - a^2 or a^2 = 130 - 49 = 81 making a = 9 = BC.
To find the exact length of BC, we can use the concept of the median in a triangle. The median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In this case, the median AM connects vertex A to the midpoint of side BC.
Let's denote the length of BC as x. Since AM is the median, its length is given as 3.5 inches.
According to the properties of a median, it divides the triangle into two smaller congruent triangles. In this case, we have two triangles: triangle ABM and triangle ACM.
In triangle ABM, we have side AB with a length of 4 inches and median AM with a length of 3.5 inches. To find the length of BM, we can use the median formula, which states:
BM = 2 * AM
Substituting the given values, we have:
BM = 2 * 3.5
BM = 7 inches
Now, let's move on to triangle ACM. We have side AC with a length of 7 inches and median AM with a length of 3.5 inches. To find the length of CM, we again use the median formula:
CM = 2 * AM
Substituting the given values, we have:
CM = 2 * 3.5
CM = 7 inches
Since BC is the sum of BM and CM, we can add the lengths of BM and CM to find the exact length of BC:
BC = BM + CM
BC = 7 + 7
BC = 14 inches
Therefore, the exact length of BC is 14 inches.