Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.
To do:
We have to prove that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.
Solution:
Let $A(x_1,y_1), B(x_2,y_2), C(x_3,y_3)$ be the vertices of a $\triangle ABC$.
Let $D$ and $E$ be the mid-points of the sides $AB$ and $AC$ respectively.
This implies,
\( \mathrm{DE}=\frac{1}{2} \mathrm{BC} \)
The coordinates of \( \mathrm{D} \) are \( \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) \)
The coordinates of \( \mathrm{E} \) are \( \left(\frac{x_{1}+x_{3}}{2}, \frac{y_{1}+y_{3}}{2}\right) \)
The length of the side $BC=\sqrt{(x_3-x_2)^2+(y_3-y_2)^2}$......(i)
The length of the side $DE=\sqrt{\left(\frac{x_{1} +x_{3}}{2} -\frac{x_{1} +x_{2}}{2}\right)^{2} +\left(\frac{y_{1} +y_{3}}{2} -\frac{y_{1} +y_{3}}{2}\right)^{2}}$
$=\sqrt{\frac{(x_1+x_3-x_1-x_2)^2}{4}+\frac{(y_1+y_3-y_1-y_2)^2}{4}}$
$=\sqrt{\frac{( x_{3} -x_{2})}{4}^{2} +\frac{( y_{3} -y_{2})}{4}^{2}}$
$=\frac{1}{2}\sqrt{(x_3-x_2)^2+(y_3-y_2)^2}$
$=\frac{1}{2}BC$ (From (i))
Hence proved.
Related Articles
- Using converse of B.P.T., prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.
- Prove that the line segment joining the mid-point of the hypotenuse of a right triangle to its opposite vertex is half of the hypotenuse.
- Circles are described on the sides of a triangle as diameters. Prove that the circles on any two sides intersect each other on the third side (or third side produced).
- Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another.
- Using B.P.T., prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.
- Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio.
- Prove that the line segment joining the point of contact of two parallel tangents of a circle passes through its center.
- The perimeter of an isosceles triangle is 50 cm. If one of the two equal sides is 18 cm, find the third side.
- Two sides of a triangle are 12cm and 14cm. The perimeter of the triangle is 36cm. What is its third side?
- If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
- Show that the mid-point of the line segment joining the points $(5, 7)$ and $(3, 9)$ is also the mid-point of the line segment joining the points $(8, 6)$ and $(0, 10)$.
- Prove that in a right angle triangle, the square of the hypotenuse is equal the sum of squares of the other two sides.
- Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.
- Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.
- Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal.
Kickstart Your Career
Get certified by completing the course
Get Started