Advertisements
Advertisements
प्रश्न
The bisectors of ∠B and ∠C of a quadrilateral ABCD intersect in P. Show that P is equidistant from the opposite sides AB and CD.
Advertisements
उत्तर
Given: A quadrilateral ABCD in which bisectors of ∠B and ∠C meet in P. PM ⊥ ABand PN ⊥ CD.
To prove: PM = PN
Construction: Draw PL ⊥ BC
Proof: Since, P lies on the bisector of ∠B
∴ P is equidistant from BC and BA
⇒ PL = PM ...(i)
Also, P lies on the bisector of ∠C ...[Given]
∴ P is equidistant from CB and CD
⇒ PL = PN ...(ii)
From (i) and (ii), we have
PL = PM
and PL = PN
⇒ PM = PN.
Hence proved.
संबंधित प्रश्न
Construct a triangle ABC, in which AB = 4.2 cm, BC = 6.3 cm and AC = 5 cm. Draw perpendicular bisector of BC which meets AC at point D. Prove that D is equidistant from B and C.
Construct a right angled triangle PQR, in which ∠Q = 90°, hypotenuse PR = 8 cm and QR = 4.5 cm. Draw bisector of angle PQR and let it meets PR at point T. Prove that T is equidistant from PQ and QR.
In the given triangle ABC, find a point P equidistant from AB and AC; and also equidistant from B and C.
Describe the locus of points at a distance 2 cm from a fixed line.
Describe the locus of the moving end of the minute hand of a clock.
Describe the locus of the door handle, as the door opens.
Describe the locus of points inside a circle and equidistant from two fixed points on the circumference of the circle.
Describe the locus of the centres of all circles passing through two fixed points.
By actual drawing obtain the points equidistant from lines m and n; and 6 cm from a point P, where P is 2 cm above m, m is parallel to n and m is 6 cm above n.
Use ruler and compasses for the following question taking a scale of 10 m = 1 cm. A park in a city is bounded by straight fences AB, BC, CD and DA. Given that AB = 50 m, BC = 63 m, ∠ABC = 75°. D is a point equidistant from the fences AB and BC. If ∠BAD = 90°, construct the outline of the park ABCD. Also locate a point P on the line BD for the flag post which is equidistant from the corners of the park A and B.
