Advertisements
Advertisements
Question
P and Q are points on the sides AB and AC respectively of a ΔABC. If AP = 2cm, PB = 4cm, AQ = 3cm and QC = 6cm, show that BC = 3PQ.
Solution
We have : `(AP)/(AB)=2/6=1/3` and `(AQ)/(AC)=3/9=1/3`
⟹ `(AP)/(AB)=(AQ)/(AC)`
In Δ APQ and Δ ABC, we have:
`(AP)/(AB)=(AQ)/(AC)`
∠𝐴= ∠𝐴
Therefore, by AA similarity theorem, we get:
Δ APQ - Δ ABC
Hence,` (PQ)/(BC)=(AQ)/(AC)=1/3`
⇒` (PQ)/(BC)=1/3`
⟹ BC = 3PQ
This completes the proof.
APPEARS IN
RELATED QUESTIONS
In figure, find ∠L
In figure, QA and PB are perpendicular to AB. If AO = 10 cm, BO = 6 cm and PB = 9 cm. Find AQ
E and F are points on the sides PQ and PR, respectively, of a ΔPQR. For the following case, state whether EF || QR.
PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
In the given figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that
(i) ΔPAC ∼ ΔPDB
(ii) PA.PB = PC.PD
State, true or false:
All isosceles triangles are similar.
ABC is a triangle. PQ is a line segment intersecting AB in P and AC in Q such that PQ || BC and divides triangle ABC into two parts equal in area. Find the value of ratio BP : AB.
A model of a ship if made to a scale of 1 : 200.
(i) Thelength of the model is 4 m; calculate the length of the ship.
(ii) The area of the deck of the ship is 160000 m2; find the area of the deck of the model.
(iii) The volume of the model is 200 litres; calculate the volume of the ship in m3.
ΔABC~ΔDEF and their areas are respectively 64 cm2 and 121cm2. If EF = 15.4cm, find BC.
State the AA-similarity criterion
In ΔPQR, PQ = 10 cm, QR = 12cm, PR = 8 cm, find the biggest and the smallest angle of the triangle.
O is any point in the interior of ΔABC. Bisectors of ∠AOB, ∠BOC and ∠AOC intersect side AB, side BC, side AC in
F, D and E respectively.
Prove that
BF × AE × CD = AF × CE × BD
Δ ABC - Δ XYZ. If area of Δ ABC is 9cm2 and area of Δ XYZ is 16cm2 and if BC= 2.1cm, find the length of YZ.
In Δ ABC, D and E are points on the sides AB and AC respectively. If AD= 4cm, DB=4.Scm, AE=6.4cm and EC=7.2cm, find if DE is parallel to BC or not.
A model of a ship is made with a scale factor of 1 : 500. Find
The volume of the ship, if the volume of its model is 200 cm3.
An aeroplane is 30m long and its model is l5 cm long. If the total outer surface area of the model is 150 cm2 , find the cost of painting the outer surface of the aeroplane at Rs. 120 per m2, if 5O m2 is left out for windows.
Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b.
D and E are points on the sides AB and AC respectively of a Δ ABC such that DE | | BC and divides Δ ABC into two parts, equal in area. Find `"BD"/"AB"`.
The sides PQ and PR of the ΔPQR are produced to S and T respectively. ST is drawn parallel to QR and PQ: PS = 3:4. If PT = 9.6 cm, find PR. If 'p' be the length of the perpendicular from P to QR, find the length of the perpendicular from P to ST in terms of 'p'.
Given that ΔABC ∼ ΔPRQ, name the corresponding angles and the corresponding sides.
Find the scale factor in each of the following and state the type of size transformation:
Image length = 6cm, Actual length = 4cm.
A plot of land of area 20km2 is represented on the map with a scale factor of 1:200000. Find: The number of KM represented by 2cm on the map.
Check whether the triangles are similar and find the value of x
In the given figure, UB || AT and CU ≡ CB Prove that ΔCUB ~ ΔCAT and hence ΔCAT is isosceles.
A flag pole 15 m high casts a shadow of 3 m at 10 a.m. The shadow cast by a building at the same time is 18.6 m. The height of the building is
Observe the figure and complete the following activity
In fig, ∠B = 75°, ∠D = 75°
∠B ≅ [ ______ ] ...[each of 75°]
∠C ≅ ∠C ...[ ______ ]
ΔABC ~ Δ [ ______ ] ...[ ______ similarity test]
Side of equilateral triangle PQR is 8 cm then find the area of triangle whose side is half of the side of triangle PQR.
In the given figure, ∠ACB = ∠CDA, AC = 8cm, AD = 3cm, then BD is ______.
If ΔABC ∼ ΔDEF and ∠A = 48°, then ∠D = ______.
