in a triangle ABC it is known that AB=AC. Suppose D is the mid point of AC and BD=BC=2. Then the area of the triangle ABC is
let AD = DC = x
then AB = 2x
in triangle ABC , angle B = angle C =Ø
in triangle BCD , angle C = angle BDC = Ø
2 angles of one are equal to 2 angles of the other, so they are similar
then:
2/x = 2x/2
2x^2 = 4
x^2=2
x=√2
So triangle ABC has equal sides of 2√2 and a base of 2
which is similar to
√2 : √2 : 1
Using the cosine law we can find angle A
1^2 = √2^2 + √2^2 - 2(√2)(√2)cosA
4cosA = 3
cosA = 3/4
then sinA = √7/4
area ABC = (1/2)(2e√2)(2√2)(√7/4)= √7
check my arithmetic, should have written it out first.
To find the area of triangle ABC, we can use the formula for the area of a triangle:
Area = (base * height) / 2
In this case, we need to find the height of triangle ABC. Let's call the point where BD intersects AC as E. We can use the fact that D is the midpoint of AC to determine the length of DE.
Since D is the midpoint of AC, we can conclude that AD = DC. We are given that BD = BC = 2. Using this information, we can calculate the length of AE as follows:
AE = AC - CE = AC - BC = AB - BC = AB - BD = AD
Now, we know that AE = AD. Triangles AED and BCD are congruent by side-side-side (SSS) congruence.
Since AE = AD, angle ADE is equal to angle EDA. Similarly, angle CBD is equal to angle BDC by congruence. Therefore, triangles ADE and BCD are also congruent by angle-angle-side (AAS) congruence.
As a consequence of the congruence, we can conclude that DE = BD = 2. We know that the base of the triangle, AC, is equal to 2*DE = 2*2 = 4.
Therefore, we have the base (AC) and the height (AE). Plugging these values into the area formula, we can find the area of triangle ABC:
Area = (base * height) / 2 = (4 * AE) / 2 = 2 * AE
Since AE = AD, the area of triangle ABC is equal to 2 * AD.
However, we currently don't have the values for AD or AB. Without additional information, we can't determine the specific area at this time.