8.2-Mid-point Theorem

8.2-Mid-point Theorem Important Formulae

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Grade 9 → Math → Quadrilaterals → 8.2-Mid-point Theorem

After successful completion of this topic, you should be able to:

  • Understand mid-point theorem.

Theorem 8.8 : The line segment joining the mid-points of two sides of a triangle is parallel to the third side.:

Theorem 8.9 : The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.:


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ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.20). AC is a diagonal. Show that :

(i) SR || AC and SR= $\dfrac{1}{2}$ AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.

ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

Solution:
Show PQRS is a parallelogram. Also show PQ || AC and PS || BD. So, ∠ P = 90°.

Suggest improvements/Report errors

ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig. 8.21). Show that F is the mid-point of BC.

In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see Fig. 8.22). Show that the line segments AF and EC trisect the diagonal BD.

Solution:
AECF is a parallelogram. So, AF || CE, etc.

Suggest improvements/Report errors

ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that:

(i) D is the mid-point of AC.
(ii) MD⊥AC
(iii) CM = MA = $\dfrac{1}{2}$ AB