Advertisements
Advertisements
Question
In the given figure, ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1cm, PB = 3cm, AQ = 1.5cm, QC = 4.5cm, prove that area of ΔAPQ is 116 of the area of ΔABC.
Solution
We have :
`(AP)/(AB)=1/1+3=1/4` amd `(AQ)/(AC)=1.5/
(1.5+4.5)=1.5/6=1/4`
⇒ `(AP)/(AB)=(AQ)/(AC)`
Also, ∠𝐴= ∠𝐴
By SAS similarity, we can conclude that ΔAPQ- ΔABC.
`(ar(Δ APQ))/(ar(Δ ABC))=(AP^2)/(AB^2)=1^2/4^2=1/16`
⇒ `(ar(ΔAPQ))/(ar(ΔABC))=1/16`
⇒` ar (ΔAPQ)=1/16xxar(ΔABC) `
Hence proved.
APPEARS IN
RELATED QUESTIONS
In figure, ∠CAB = 90º and AD ⊥ BC. If AC = 75 cm, AB = 1 m and BD = 1.25 m, find AD.
In figure, considering triangles BEP and CPD, prove that BP × PD = EP × PC.
A vertical stick 20 cm long casts a shadow 6 cm long on the ground. At the same time, a tower casts a shadow 15 m long on the ground. Find the height of the tower.
In the adjoining figure, ABC is a right angled triangle with ∠BAC = 90°.
1) Prove ΔADB ~ ΔCDA.
2) If BD = 18 cm CD = 8 cm Find AD.
3) Find the ratio of the area of ΔADB is to an area of ΔCDA.
State, true or false:
Two isosceles-right triangles are similar.
Given: FB = FD, AE ⊥ FD and FC ⊥ AD.
Prove that: `(FB)/(AD) = (BC)/(ED)`.
The given figure shows a trapezium in which AB is parallel to DC and diagonals AC and BD intersect at point P. If AP : CP = 3 : 5,
Find:
- ∆APB : ∆CPB
- ∆DPC : ∆APB
- ∆ADP : ∆APB
- ∆APB : ∆ADB
On a map, drawn to a scale of 1 : 250000, a triangular plot PQR of land has the following measurements :
PQ = 3cm, QR = 4 cm and angles PQR = 90°
(i) the actual lengths of QR and PR in kilometer.
(ii) the actual area of the plot in sq . km.
In the given figure, DE║BC. If DE = 3cm, BC = 6cm and ar(ΔADE) = `15cm^2`, find the area of ΔABC.
Figure shows Δ PQR in which ST || QR and SR and QT intersect each other at M. If `"PT"/"TR" = 5/3` find `("Ar" (triangle "MTS"))/("Ar" (triangle "MQR"))`
In figure , DEF is a right -angled triangle with ∠ E = 90 °.FE is produced to G and GH is drawn perpendicular to DE = 8 cm , DH = 8 cm ,DH = 6 cm and HF = 4 cm , find `("Ar" triangle "DEF")/("Ar" triangle "GHF")`
Δ ABC is similar to Δ PQR. If AB = 6cm, BC = 9cm, PQ = 9cm and PR = 10.5cm, find the lengths of AC and QR.
A model of a ship is made to a scale 1 : 300.
- The length of the model of the ship is 2 m. Calculate the length of the ship.
- The area of the deck of the ship is 180,000 m2. Calculate the area of the deck of the model.
- The volume of the model is 6.5 m3. Calculate the volume of the ship.
In the given figure, PQ || AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :
- `(CP)/(PA)`
- PQ
- If AP = x, then the value of AC in terms of x.
In the figure, given below, AB, CD and EF are parallel lines. Given AB = 7.5 cm, DC = y cm, EF = 4.5 cm, BC = x cm and CE = 3 cm, calculate the values of x and y.
In the following figure, M is mid-point of BC of a parallelogram ABCD. DM intersects the diagonal AC at P and AB produced at E. Prove that : PE = 2 PD
The given figure shows a parallelogram ABCD. E is a point in AD and CE produced meets BA produced at point F. If AE = 4 cm, AF = 8 cm and AB = 12 cm, find the perimeter of the parallelogram ABCD.
In fig.DE || BC ,AD = 1 cm and BD = 2 cm. what is the ratio of the ar(ΔABC) to the ar (ΔADE)?
If ΔABC ~ ΔDEF, then writes the corresponding congruent angles and also write the ratio of corresponding sides.
Construct a triangle ABC with side BC = 6 cm, ∠B = 45°, ∠A = 105°. Then construct another triangle whose sides are `(3)/(4)` times the corresponding sides of the ΔABC.
Find the scale factor in each of the following and state the type of size transformation:
Actual length = 12cm, Image length = 15cm.
Find the scale factor in each of the following and state the type of size transformation:
Model area = 75cm2, Actual area = 3cm2
A map is drawn to scale of 1:20000. Find: The distance on the map representing 4km
In the adjacent figure, ∆ABC is right angled at C and DE ⊥ AB. Prove that ∆ABC ~ ∆ADE and hence find the lengths of AE and DE
A man whose eye-level is 2 m above the ground wishes to find the height of a tree. He places a mirror horizontally on the ground 20 m from the tree and finds that if he stands at a point C which is 4 m from the mirror B, he can see the reflection of the top of the tree. How height is the tree?
From the given figure, prove that ΔABC ~ ΔEDF
Given ΔABC ~ ΔDEF, if ∠A = 45° and ∠E = 35° then ∠B = ?
Observe the figure and complete the following activity
In fig, ∠B = 75°, ∠D = 75°
∠B ≅ [ ______ ] ...[each of 75°]
∠C ≅ ∠C ...[ ______ ]
ΔABC ~ Δ [ ______ ] ...[ ______ similarity test]
In Quadrilateral ABCD, side AD || BC, diagonal AC and BD intersect in point P, then prove that `"AP"/"PD" = "PC"/"BP"`
In ΔABC, PQ || BC. If PB = 6 cm, AP = 4 cm, AQ = 8 cm, find the length of AC.
