Advertisements
Advertisements
प्रश्न
A tangent ADB is drawn to a circle at D whose centre is C. Also, PQ is a chord parallel to AB and ∠QDB = 50°. Find the value of ∠PDQ.
उत्तर
Given: Chord PQ || tangent AB
∴ ∠Q = ∠QDB = 50° ......[Alternate interior angles]
And ∠CDB = ∠PRD ......[Alternate interior angles]
ADB is a tangent at point D and CD is the circle's radius.
Thus, ∠CDB = 90° ......[Tangent theorem]
⇒ ∠CDQ + ∠QDB = 90°
⇒ ∠CDQ + 50° = 90°
⇒ ∠CDQ = 90° – 50° = 40°
CR is a perpendicular to the chord PQ, so it bisects PQ.
Thus, ∠PRC = 90° and PR = QR
Now, in ΔDRP and ΔDRQ
PR ≅ QR .....[Proved above]
∠DRP ≅ ∠DRQ ......[Each 90°]
DR ≅ DR ......[Common side]
Thus ΔDRP ≅ ΔDRQ .....[By SAS criterion of congruence)
So, ∠P ≅ ∠Q .....[C.A.C.T.]
⇒ ∠P = 50° .....[∵ ∠Q = 50°]
Now, the sum of the interior angles in a triangle is 180°.
Thus, ∠P + ∠PRD + ∠RDP = 180°
50° + 90° + ∠RDP = 180°
∠RDP = 180° – 50° – 90° = 40°
Now, ∠PDQ = ∠RDP + ∠RDQ
⇒ ∠PDQ = 40° + 40° = 80°
Hence, ∠PDQ = 80°.
APPEARS IN
संबंधित प्रश्न
State which pair of triangles in the given figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:
State which pair of triangles in the following figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:
In the following figure, if ΔABE ≅ ΔACD, show that ΔADE ∼ ΔABC.
A vertical pole of a length 6 m casts a shadow 4m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
In the following figure, AB || QR. Find the length of PB.
In the following figure, PA, QB and RC are each perpendicular to AC. Prove that `1/x+1/z=1/y`
Two triangles DEF an GHK are such that ∠D = 48° and ∠H = 57° . If ΔDEF ∼GHK then find the measures of ∠F
In figure, BD and CE intersect each other at the point P. Is ΔPBC ~ ΔPDE? Why?
In triangle PQR and MST, ∠P = 55°, ∠Q = 25°, ∠M = 100° and ∠S = 25°. Is ∆QPR ~ ∆TSM? Why?
In the figure, if ∠ACB = ∠CDA, AC = 8 cm and AD = 3 cm, find BD.
A 15 metres high tower casts a shadow 24 metres long at a certain time and at the same time, a telephone pole casts a shadow 16 metres long. Find the height of the telephone pole.
A street light bulb is fixed on a pole 6 m above the level of the street. If a woman of height 1.5 m casts a shadow of 3 m, find how far she is away from the base of the pole.
If in two triangles ABC and PQR, `(AB)/(QR) = (BC)/(PR) = (CA)/(PQ)`, then ______.
If in triangles ABC and DEF, `(AB)/(DE) = (BC)/(FD)`, then they will be similar, when ______.
Is it true to say that if in two triangles, an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle, then the triangles are similar? Give reasons for your answer.
In the figure with ΔABC, P, Q, R are the mid-points of AB, AC and BC respectively. Then prove that the four triangles formed are congruent to each other.
In the given figure below, `(AD)/(AE) = (AC)/(BD)` and ∠1 = ∠2, Show that ΔBAE ∼ ΔCAD.
Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ = 3 RS. Find the ratio of the areas of triangles POQ and ROS.
