Advertisements
Advertisements
प्रश्न
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
- D is the mid-point of AC
- MD ⊥ AC
- CM = MA = `1/2AB`
Advertisements
उत्तर
(i) In ΔABC,
It is given that M is the mid-point of AB and MD || BC.
Therefore, D is the mid-point of AC. ...(Converse of mid-point theorem)
(ii) As DM || CB and AC is a transversal line for them, therefore,
∠MDC + ∠DCB = 180° ...(Co-interior angles)
∠MDC + 90° = 180°
∠MDC = 90°
∴ MD ⊥ AC
(iii) Join MC.
In ΔAMD and ΔCMD,
AD = CD ...(D is the mid-point of side AC)
∠ADM = ∠CDM ...(Each 90º)
DM = DM (Common)
∴ ΔAMD ≅ ΔCMD ...(By SAS congruence rule)
Therefore, AM = CM ...(By CPCT)
However, AM = `1/2AB` ...(M is the mid-point of AB)
Therefore, it can be said that
CM = AM = `1/2AB`
APPEARS IN
संबंधित प्रश्न
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
In a triangle ∠ABC, ∠A = 50°, ∠B = 60° and ∠C = 70°. Find the measures of the angles of
the triangle formed by joining the mid-points of the sides of this triangle.
Fill in the blank to make the following statement correct
The triangle formed by joining the mid-points of the sides of an isosceles triangle is
Fill in the blank to make the following statement correct:
The figure formed by joining the mid-points of consecutive sides of a quadrilateral is
In triangle ABC ; D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F.
Prove that BDEF is a parallelogram. If AB = 16 cm, AC = 12 cm and BC = 18 cm,
find the perimeter of the parallelogram BDEF.
In a right-angled triangle ABC. ∠ABC = 90° and D is the midpoint of AC. Prove that BD = `(1)/(2)"AC"`.
In ΔABC, the medians BE and CD are produced to the points P and Q respectively such that BE = EP and CD = DQ. Prove that: A is the mid-point of PQ.
The diagonals AC and BD of a quadrilateral ABCD intersect at right angles. Prove that the quadrilateral formed by joining the midpoints of quadrilateral ABCD is a rectangle.
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if ______.
Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.
