Find The Relation Between The Area Of Two Triangles
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How do find the relation between the area of the blue triangle and the area of the red triangle?
![Geometry math problem Find The Relation Between The Area Of Two Triangles](https://aplusreach.com/wp-content/uploads/2022/03/2-1024x683.jpg)
From figure, MNO is a triangle, MP = 3 cm, PQ = 4 cm, QN = 5 cm, MR = 5 cm & RO = 4 cm. Then find the relation between the area of two triangles
Solution
We can solve this geometry math problem in different ways, here I am presenting two methods
Method 1
![Area of triangle using sine rule](https://aplusreach.com/wp-content/uploads/2022/03/3-1024x683.jpg)
Assume angle OMN is θ and area of four triangles are A, B, C and D, so
Area of triangle PMR = ½ × MP × MR × sin θ
⟹ Area of triangle PMR = ½ × 3 × 5 × sin θ
⟹ A = (15 sin θ)/2……………………………………………..eq(1)
Area of triangle MQR = ½ × MQ × MR × sin θ
⟹ Area of triangle MQR = ½ × 7 × 5 × sin θ
⟹ A + B = (35 sin θ)/2……………………………………………..eq(2)
Area of triangle MQO = ½ × MQ × MO × sin θ
⟹ Area of triangle MQO = ½ × 7 × 9 × sin θ
⟹ A + B + C = (63 sin θ)/2……………………………………………..eq(3)
Area of triangle MNO = ½ × MN × MO × sin θ
⟹ Area of triangle MNO = ½ × 12 × 9 × sin θ
⟹ A + B + C + D = (108 sin θ)/2……………………………………………..eq(4)
Subtract equation 3 from equation 4, then
A + B + C + D – (A + B + C)= (108 sin θ)/2 – (63 sin θ)/2
⟹ 2D = 45 sin θ
Subtract equation 1 from equation 2, then
A + B – A = (35 sin θ)/2 – (15 sin θ)/2
⟹ B = (20 sin θ)/2
⟹ sin θ = B / 10
2D = 45 sin θ = 45 (B/10)
⟹ 20D = 45B
⟹ 4D = 9B
So, 4 × Area of the red triangle = 9 × Area of the blue triangle
Method 2
Before moving to 2nd method we need to discuss the equation to find the relation between the area of triangles
Relation between the area of triangle and height
![relation between the area of triangle and height](https://aplusreach.com/wp-content/uploads/2022/03/relation-between-the-area-of-triangle-and-height-1024x683.jpg)
From the figure, area of red triangle = ½ × a × h
area of blue triangle = ½ × b × h
⟹ h/2 = area of red triangle/a = area of blue triangle/b
If the height of the triangle is equal, then the area of the triangle divided with base length then we get a constant
Now solving for the relation between the area of triangles
![](https://aplusreach.com/wp-content/uploads/2022/03/Relation-between-the-area-of-triangle-and-height-1-1024x683.jpg)
From the figure triangle MPR and triangle QPR has the same height, so
A/3 = B/4
⟹ B = 4A/3
Triangle MRQ and triangle ORQ has the same height, so
(A + B)/5 = C/4
⟹ A + B = 5C/4
⟹ A + 4A/3 = 5C/4
Thus, 7A/3 = 5C/4
⟹ C = 28A/15
Triangle MQO and triangle QNO has the same height, then
(A + B + C)/7 = D/5
⟹ D/5 = (A + 4A/3 + 28A/15)/7
⟹ D/5 = A(7/3 + 28/15)/7
So, D = 5(1/3 + 4/15)A
⟹ D = 3A
Then, B/D = (4A/3)/(3A)
⟹ B/D = 4/9
⟹ 9B = 4D
So, 4 × Area of the red triangle = 9 × Area of the blue triangle