Advertisements
Advertisements
Question
In a circle, two chords AB and CD intersect at a point P inside the circle. Prove that
(a) ΔPAC ∼PDB (b) PA. PB= PC.PD
Solution
Given : AB and CD are two chords
To Prove:
(a) Δ PAC ~ ΔPDB
(b) PA.PB = PC.PD
Proof: In Δ PAC and Δ PDB
∠𝐴𝑃𝐶 = ∠𝐷𝑃𝐵 (𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠)
∠𝐶𝐴𝑃 = ∠𝐵𝐷𝑃 (𝐴𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙)
𝑏𝑦 𝐴𝐴 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑜𝑛 Δ𝑃𝐴𝐶 ~𝑃𝐷𝐵
When two triangles are similar, then the ratios of lengths of their corresponding sides are proportional.
`∴ (PA)/(PD)=(PC)/(PB)`
⟹ PA. PB = PC. PD
APPEARS IN
RELATED QUESTIONS
In figure, ∠A = ∠CED, prove that ∆CAB ~ ∆CED. Also, find the value of x.
Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
State, true or false:
Two congruent polygons are necessarily similar.
In the given figure, ∠ABC = 90° and BD⊥AC. If AB = 5.7cm, BD = 3.8cm and CD = 5.4cm, find BC.
In the given figure, ∠ABC = 90° and BD⊥AC. If BD = 8cm, AD = 4cm, find CD.
In an isosceles ΔABC, the base AB is produced both ways in P and Q such that
AP × BQ = AC2.
Prove that ΔACP~ΔBCQ.
The areas of two similar triangles ABC and PQR are in the ratio 9:16. If BC = 4.5cm, find the length of QR.
In the given figure, ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and the distance between AB and AC is 14 cm. If arcs of equal radii 7 cm taking A, B, C and D as centres, have been drawn, then find the area of the shaded region ?
In Δ ABC , DE is parallel to BC and `"AD"/"DB" = 2/7` IF AC = 5 .6 , find AE.
AD and BC are two straight lines intersecting at 0. CD and BA are perpendirulars from Band Con AD. If AB=6cm, CD =9cm, AD =20cm and BC=25cm, find the lengths of AO, BO, CO and DO.
D and E are points on the sides AB and AC respectively of Δ ABC such that AB=5.6cm, AD= 1.4cm, AC=7 .2cm and AE = 1.5 cm, show that DE is parallel to BC
A model of a ship is made with a scale factor of 1 : 500. Find
The deck area of the model, if the deck area of the ship is 1500000 m2
The actual area of an island is 1872km2 . On a map, this area is 117 cm2. if the length of the coastline is 44cm on the map, find the length of its actual coastline.
In the following figure, in Δ PQR, seg RS is the bisector of ∠PRQ.
PS = 11, SQ = 12, PR = 22. Find QR.
In the following figure, point D divides AB in the ratio 3 : 5. Find :
BC = 4.8 cm, find the length of DE.
If ΔABC, D and E are points on AB and AC. Show that DE || BC for each of the following case or not:
AB = 5.6cm, AD = 1.4cm, AC = 7.2cm, and AE = 1.8cm
In figure, PQ is parallel to BC, AP : AB = 2 : 7. If QC = 0 and BC = 21,
Find
(i) AQ
(ii) PQ
In the figure, PQR is a straight line and PS || RT. If QS = 12cm, QR = 15cm, QT = 10cm and RT = 6cm, find PQ and PS.
In the figure, AB || RQ and BC || SQ, prove that `"PC"/"PS" = "PA"/"PR"`.
D and E are points on the sides AB and AC of ΔABC such that DE | | BC and divides ΔABC into two parts, equal in area. Find `"BD"/"AB"`.
On a map drawn to a scale of 1: 2,50,000, a triangular plot of land has the following measurements:
AB = 3 cm, BC = 4 cm, ∠ABC = 90°. Calculate:
(i) The actual length of AB in km.
(ii) The area of Plot in sq. km.
A map is drawn to scale of 1:20000. Find: The area of the lake on the map which has an actual area of 12km2
A model of cargo tuck is made to a scale of 1:40. The length of the model is 15cm. Calculate: The volume of the model if the volume of the truck is 6m3
In ∆LMN, ∠L = 60°, ∠M = 50°. If ∆LMN ~ ∆PQR then the value of ∠R is
PR = 26 cm, QR = 24 cm, ∠PAQ = 90°, PA = 6 cm and QA = 8 cm. Find ∠PQR
In fig., seg AC and seg BD intersect each other at point P.
`"AP"/"PC" = "BP"/"PD"` then prove that ΔABP ~ ΔCDP
Is the following statement true? Why? “Two quadrilaterals are similar, if their corresponding angles are equal”.
