This question was uploaded on 23/08/21 on social media accounts.
A square and one of its diagonal is drawn. One side of square is divided in three equal parts and two lines are drawn from joining of each part to the vertices of square (Except vertices include in diagonal). Find the area of the triangle formed between diagonal and two lines.
Step 1:
Give names for every vertices and intersection of lines.
Step 2:
Observe △APD and △CPM,
Both are similar.
△APD and~ △CPM
⇒ PD : PM = AD : CM
⇒ PD : PM = 6 : 4
Let, PD = 6a and PM = 4a
Observe △AQD and △CQN,
Both are similar.
△AQD and~ △CQN
⇒ QD : QN = AD : CN
⇒ QD : QN = 6 : 2
Let, QD = 6b and QN = 2b
Step 3:
Let angle between DM and DN = 'x'
A(MND) = MN × CD / 2 = 2 × 6 / 2 = 6
Also,
A(MND) = (10a) × (8b) × sin(x) / 2
⇒ 6 = 40ab sin(x)
⇒ ab sin(x) = 3 / 20
A(PQD) = (6a) × (6b) × sin(x) / 2 = 18ab sin(x)
⇒ A(PQD) = 18 × 3 / 20
⇒ A(PQD) = 27 / 10 = 2.7
Puzzle related to Geometry, Square, Area
