Rachit is having an open space behind his house. He plans to make a garden inthis land. He divided the land into 4 parts and decided to plant the rose plants,grass, vegetables and some trees as shown in the figure.
(a) If PQ || RS, RP= x –3 , AR = 2x, SQ = x –2 and AS = 2x + 3 , find the value of x.(1)(i) 9 (ii)19 (iii)29 (iv) 16
(b) If RS || DC, which of the following statements is true?(1)(i) APQ~∆ABC(ii) ∆ARS~∆ABC (iii) ∆ARS~∆ADC (iv) ∆ADC~∆ABC
4
(c) If ∆APQ~∆ADCsuch that PQ = 3 m, DC = 4 m and area of ∆ADC is 96 m2. Find the area he used to plant the roses.(1)(i) 72 m2(ii)84 m2(iii) 52 m2(iv) 54 m2
(d) Rachit wanted to differentiate the vegetable garden from the rest of his garden. So he laid small pebbles along the length AC. If AB is perpendicular to BC and AB= 5 m ,BC = 12 m , find the length of AC.(1)(i) 11 m(ii) 13 m(iii) 15 m(iv) 16 m
Answer fast plzzzz
Answers
a) Answer:
Between the triangle ARP and CRQ applying mid point theorem
RP ∥ BC and
RP =
2
1
BC = CQ.
And AR = RC ( R is the mid point of AC )
again PR ∥ BC and AC is the transversal.
Therefore angle ARP = angle RCQ.
Therefore the triangles are congruent by SAS test.
Area ΔARP=AreaΔ RCQ.
By applying the same midpoint theorem we can prove that each of the four triangles have the same area.
So, they divide the triangle into four equal areas.
Now total area = 20 sq. cm.
Therefore area of the Δ PQR is 20 sq.cm divided by 4 = 5 sq.cm
b) Refer the attachment friend ↗️↗️
c) ANSWER
ΔABC∼ΔAPQ (Given)
∴
ar(ΔAPQ)
ar(ΔABC)
=
PQ
2
BC
2
...(i)
[∴ Ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides]
But ar(ΔAPQ)=4ar(ΔABC) (given)
∴
ar(ΔAPQ)
ar(ΔABC)
=
4
1
[From (i) and (ii)
∴ BC/PQ= 1/2
Step-by-step explanation:
a. 9
PR/AR=QS/AS ( B.P.T)
x-3/2x=x-2/2x+3
x=9
b. triangleARS similar triangleABC
c. triangle APQ similar triangle ADC
hence, ar of APQ/ar of ADC = side of triangle APQ ^2/ side of triangle ADC ^2 =ar(APQ)/96=3^2/4^2
ar(APQ)=96×9/16=54m^2
option iv
d. AB perpendicular to BC
using Pythagoras theorem we get
AC^2=12^2+5^2
AC^2=169
AC=13m
option ii