Point D is the midpoint of median AM of triangle ABC. Point E is the midpoint of AB, and point T is the intersection of BD and ME. Find the area of triangle DMT if ABC =150.
Please stop cheating on challenge problems, or we will have to deactivate your account without refunds.
yep
but howd you know their account from just their nickname lol
We can though...
To find the area of triangle DMT, we need to obtain the lengths of its sides.
Given that point D is the midpoint of median AM in triangle ABC, we know that AD = DM.
Additionally, we know that point E is the midpoint of side AB, so AE = EB.
Since point T is the intersection of BD and ME, we can see that triangle BTE is similar to triangle DTA. This is because both triangles share angle T, and angles BTE and DTA are both right angles (as BD and ME are perpendicular bisectors in right triangles BDM and AEM, respectively).
From the similarity of triangles BTE and DTA, we can write the following proportion:
BT / DT = TE / TA
Since AE = EB, we have TE = 3 / 2 * AE = 3 / 2 * EB.
Since AD = DM, we have DT = DT = 1 / 2 * AM.
Substituting the above values, we get the equation:
BT / (1 / 2 * AM) = (3 / 2 * EB) / AM
Simplifying, we get:
BT = (3 / 2 * EB) / (1 / 2)
BT = 3 * EB.
Since triangle ABC is given to be 150, we can express EB in terms of AB. Since E is the midpoint of AB, we know that AE = EB = AB / 2.
Substituting this into the expression for BT, we get:
BT = 3 * EB = 3 * (AB / 2) = 3AB / 2.
Now, let's find the area of triangle DMT. We can use Heron's formula to do this.
Heron's formula states that the area of a triangle with side lengths a, b, and c is given by:
Area = √[s(s - a)(s - b)(s - c)],
where s is the semi-perimeter of the triangle, defined as:
s = (a + b + c) / 2.
In triangle DMT, we need to determine the side lengths DT, DM, and MT.
We know that DT = 1 / 2 * AM, and AD = DM.
Given that point D is the midpoint of median AM of triangle ABC, AM is equal to 2 * AD.
Substituting these values into DT and DM, we get:
DT = 1 / 2 * (2 * AD) = AD,
DM = AD.
We also know that BT = 3AB / 2.
Since point T is the intersection of BD and ME, we can find the length of MT by using the similarity of triangles BTE and DTA.
Since BT / DT = TE / TA, we can solve for TA:
TA = TE * DT / BT,
TA = (3EB) / (3AB / 2),
TA = 2EB / AB.
From triangle TAB, using the Pythagorean theorem, we know that AB^2 = AE^2 + EB^2.
Since AE = EB = AB / 2, we have:
(AB / 2)^2 = (AB / 2)^2 + EB^2.
Solving for EB^2:
EB^2 = (AB / 2)^2 - (AB / 2)^2,
EB^2 = 0.
Since EB^2 = 0, we have EB = 0. This implies that point B is coincident with point E, which means that triangle ABC does not exist.
Therefore, it is not possible to determine the area of triangle DMT as triangle ABC is not valid according to the given information.
It is possible. We can track their IP address and link it to their AoPS account.
subscribe to my youtube channel: AoPS_Fortnite
I am giving the answer to you one time, stop cheating...
The answer is 25/2
BM is parallel to AC, and 1/2 as long
so ∆DEM has 1/4 the area of ∆ABC.
See what you can do with that.