Advertisements
Advertisements
प्रश्न
In the given figure P is a midpoint of chord AB of the circle O. prove that OP ^ AB.
Advertisements
उत्तर
Given:
In the figure, O is centre of the circle and ab is chord.
P is the mid-point of AB ⇒ AP = PB
To prove: OP ⊥ AB
Construction: Join OA and OB
Proof:
In ΔOAP and ΔOBP
OA = OB ....[radii of the same circle]
OP = OP ....[common]
AP = PB ....[given]
∴ By Side-Side-Side criterion of congruency,
ΔOAP ≅ ΔOBP
The corresponding parts of the congruent triangles are congruent.
∴ ∠OPA = ∠OPB
But ∠OPA + ∠OPB = 180° ....[linear pair]
∴ ∠OPA = ∠OPB = 90°
Hence OP ⊥ AB.
APPEARS IN
संबंधित प्रश्न
Prove that the sum of three altitudes of a triangle is less than the sum of its sides.
In the given figure, AB ⊥ BE and FE ⊥ BE. If BC = DE and AB = EF, then ΔABD is congruent to
In the given figure, if AE || DC and AB = AC, the value of ∠ABD is
In the pair of triangles in the following figure, parts bearing identical marks are congruent. State the test and the correspondence of vertices by the triangle in pairs is congruent.
State, whether the pairs of triangles given in the following figures are congruent or not:
Which of the following pairs of triangles are congruent? Give reasons
ΔABC;(∠B = 70°,BC = 6cm,∠C = 50°);
ΔXYZ;(∠Z = 60°,XY = 6cm,∠X = 70°).
In the figure, RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT.
Sides, AB, BC and the median AD of ΔABC are equal to the two sides PQ, QR and the median PM of ΔPQR. Prove that ΔABC ≅ ΔPQR.
In ΔABC, AB = AC, BM and Cn are perpendiculars on AC and AB respectively. Prove that BM = CN.
∆ABC and ∆PQR are congruent under the correspondence:
ABC ↔ RQP
Write the parts of ∆ABC that correspond to
(i) `bar"PQ"`
(ii)∠Q
(iii) `bar"RP"`
