Advertisements
Advertisements
Question
In the given figure, two chords AB and CD intersect each other at the point P. prove that:
(i) ΔAPC ∼ ΔDPB
(ii) AP.BP = CP.DP
Solution
Let us join CB.
(i) In ΔAPC and ΔDPB,
∠APC = ∠DPB (Vertically opposite angles)
∠CAP = ∠BDP (Angles in the same segment for chord CB)
ΔAPC ∼ ΔDPB (By AA similarity criterion)
(ii) We have already proved that
ΔAPC ∼ ΔDPB
We know that the corresponding sides of similar triangles are proportional.
`:. (AP)/(DP) = (PC)/(PB) = (CA)/(BD)`
`=>(AP)/(DP) = (PC)/(PB)`
∴ AP. PB = PC. DP
APPEARS IN
RELATED QUESTIONS
In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:
ΔABD ∼ ΔCBE
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.
A vertical stick 10 cm long casts a shadow 8 cm long. At the same time a shadow 30 m long. Determine the height of the tower.
The sides of certain triangles are given below. Determine which of them right triangles are.
9cm, 16cm, 18cm
The sides of certain triangles are given below. Determine which of them right triangles are.
(a – 1) cm, `2sqrta` cm, (a + 1) cm
In a trapezium ABCD, it is given that AB║CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar(ΔAOB) = 84cm2. Find ar(ΔCOD).
If in two triangles DEF and PQR, ∠D = ∠Q and ∠R = ∠E, then which of the following is not true?
In triangles ABC and DEF, ∠B = ∠E, ∠F = ∠C and AB = 3DE. Then, the two triangles are ______.
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.
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.
