**Triangle worksheet solution5 :**

Here we are going to see solution for each problems of the topic area of triangle worksheet.

(5) If the three points (h, 0) (a, b) and (0, k) lie on a straight line, then using the area of the triangle formula show that (a/h) + (b/k) = 1,where h, k ≠ 0

**Solution :**

Let A(h, 0) B (a, b) and C (0, k) are the three points

Since the three points A (h, 0) B (a, b) and C (0, k) lie on a straight line we can say that the three points are collinear

So , area of triangle ABC = 0

(1/2) [ (h b + a k + 0) – (0 + 0 + kh) ] = 0

[ h b + a k – k h ] = 0 x 2

h b + a k - k h = 0

h b + a k = k h

Divided (k h) on both sides,

(h b)/(k h) + (a k)/(k h) = (k h)/(k h)

(b/k) + (a/h) = 1

(a/h) + (b/k) = 1

(7) Find the area of the triangle formed by joining the midpoints of the sides of a triangle whose vertices are (0, -1) (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.

**Solution :**

Let A (0,-1) B (2,1) and C (0,3) are the vertices of the triangle. D,E and F are the midpoints of the side AB,BC and CA respectively.

Midpoint of AB = (x_{1 }+ x_{2})/2 ,
(y_{1 }+ y_{2})/2

= (0 + 2)/2 , (-1 + 1)/2

= 2/2 , 0/2

= D (1,0)

Midpoint of BC = (x_{1 }+ x_{2})/2 , (y_{1 }+ y_{2})/2

= (2 + 0)/2 , (1 + 3)/2

= 2/2 , 4/2

= E (1,2)

Midpoint of CA = (x1+x2)/2 , (y1+y2)/2

= (0 + 0)/2 , (3 + (-1))/2

= 0/2 , 2/2

= F (0,1)

= (1/2) [(0 + 6 + 0) – (-2 + 0 + 0)]

= (1/2) [6+2]

= (1/2) [8]

= 4 square units

= (1/2) [(2 + 1 + 0) – (0 + 0 + 1)]

= (1/2) [3-1]

= (1/2) [2]

= 1 square units

Area of triangle ABC: Area of triangle DEF

4 : 1

- Section formula
- Area of quadrilateral
- Centroid
- Midpoint
- Distance between two points
- Distance between two points worksheet
- Slope of the line
- Equation of the line
- Equation of line using two points worksheet
- Equation of the line using point and slope worksheets
- Point of intersection of two lines
- concurrency of straight line
- Circumcentre of a triangle
- Orthocentre of a triangle
- Incentre of a triangle
- Locus
- Perpendicular distance
- Angle between two straight lines
- Equation of a circle
- With center and radius
- With endpoints of a diameter
- Equation of a circle passing though three points
- Length of the tangent to a circle
- Equation of the tangent to a circle
- Family of circles
- Orthogonal circles

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