Question

Find the coordinates of the mid-points of the sides of the triangle whose vertices are A(0,1), B(4,1) and C(0,7). Also find the (i) lengths of the segments joining these midpoints (ii) lengths of the sides of the triangle ABC  (iii) What relation do you observe between the lengths of these segments and the sides of the triangle

Answer 1

Given:

The vertices of the triangle are A(0,1), B(4,1) and C(0,7)

To find:

(i) The coordinates of the mid-points of the sides of the triangle

(ii) Lengths of the segments joining the midpoints

(iii) Lengths of the sides of the triangle ABC  

(iv) The relation observed between the lengths of the segments and the sides of the triangle

Answer:

(i) The coordinates of the mid-points of the sides of the triangle

• The formula for the mid-point of a side is $(\frac{x1 +x2}{2} + \frac{y1 + y2}{2})$   -(1)
• Let the three midpoints of the sides be P, Q and R respectively.

• Then, by the above formula, P = (2,1), Q = (2,4) and R = (0,4)

• The coordinates of the mid-points of the sides of the triangle are (2,1), (2,4) and (0,4).

(ii) Lengths of the segments joining the midpoints

• Length of a segment = $\sqrt{(x2-x1)^{2}+(y2 - y1)^{2} }$   -(2)
• Thus, using the above formula the lengths of the segments are calucated to be 3 units, 2 units and 5 units respectively.

• Lengths of the segments joining the midpoints are 3 units, 2 units and 5 units respectively.

(iii) Lengths of the sides of the triangle ABC

• The lengths of the sides of the triangle can be found using equation (2).
• We know that A(0,1), B(4,1) and C(0,7) are the vertices of the triangle.
• Thus the lengths of the sides are 4 units, 7.2 units and 6 units respectively.

• Lengths of the sides of the triangle ABC are 4 units, 7.2 units and 6 units respectively.

(iv) The relation observed between the lengths of the segments and the sides of the triangle

• From the lengths of the sides formed by the midpoints and the sides of the triangle it can be observed that 2 : 4 :: 3 : 6 :: 5 : 7.2 are approximately baring the same ratio of 1/2.

• The relation observed between the lengths of the segments and the sides of the triangle is that they approximately bare a constant ratio of 1/2.