Point S is on the side
PR of ∆PMR such
that 3SR = 2SP
seg ST || side PM. If
A(APMR) = 50 cm²
then find (i) A(ARST) (ii) A(PMTS). pls answer it's urgent tomorrow is my maths test
Answers
ANSWER
i) A( triangle RST) is 8 cm².
(ii) A( quadrilateral PMTS) is 42 cm².
STEP-BY-STEP EXPLANATIO
It is given that,
Point S is on the side PR of triangle PMR such that
3SR = 2SP
⇒ SR/SP = 2/3 ……. (i)
Seg ST // Seg PM
Area (triangle PMR) = 50 cm² …. (ii)
Case (i): Finding the area of triangle RST
Consider ∆RST and ∆PRM, we get
∠R = ∠R ….. [common angle]
∠RST = ∠RPM ……. [corresponding angles]
∴ By AA similarity, ∆RST ~ ∆PRM
We know that the ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides.
∴ [Area(∆RST)] / [Area(∆PRM)] = [RS²] / [PR²]
⇒ [Area(∆RST)] / [Area(∆PMR)] = [SR²] / [(SR + SP)²]
Substituting the values from (i) & (ii)
⇒ [Area(∆RST)] / [50] = [2²] / [(2 + 3)²]
⇒ [Area(∆RST)] / [50] = [4] / [25]
⇒ [Area(∆RST)] = [4/25] * 50
⇒ [Area(∆RST)] = 8 cm²
Case (ii): Finding the area of quadrilateral PMTS
The area of quadrilateral PMTS is given by,
= [Area of triangle PMR] – [Area of triangle RST]
= [50 cm²] – [8 cm²]
= 42 cm²
Step-by-step explanation:
Given:
Point S is on the side PR of ∆PMR such that 3SR=2SP.
Seg ST || seg PM
A(∆PMR)=50 cm²
To find:
(a) A(∆RST)
(b) A(□PMTS)
Solution:
We have,
3SR = 2SP
⇒ \frac{SR}{SP} = \frac{2}{3}SPSR=32 .........(1)
Finding the area of triangle RST:
Consider ∆RST and ∆PRM, we get
∠SRT = ∠PRM → [common angle]
∠RST = ∠RPM → [corresponding angles]
∴ By AA similarity → ∆RST ~ ∆PRM
We know that → the ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides.
Therefore,
\frac{Area(\triangle RST)}{Area(\triangle PRM} = \frac{(RS)^2}{(PR)^2}Area(△PRMArea(△RST)=(PR)2(RS)2
\implies \frac{Area(\triangle RST)}{Area(\triangle PMR)} = \frac{(SR)^2}{(SR + SP)^2}⟹Area(△PMR)Area(△RST)=(SR+SP)2(SR)2
On substituting the values from equation1 and A(∆PMR)=50 cm²
\implies \frac{Area(\triangle RST)}{50} = \frac{(2)^2}{(2 + 3)^2}⟹50Area(△RST)=(2+3)2(2)2
\implies Area(\triangle RST) = \frac{(2)^2\times 50}{(5)^2}⟹Area(△RST)=(5)2(2)2×50
\implies Area(\triangle RST) = \frac{4 \times 50}{25}⟹Area(△RST)=254×50
\implies \boxed{\bold{Area(\triangle RST) = 8\:cm^2}}⟹Area(△RST)=8cm2
Finding the area of quadrilateral PMTS
The area of quadrilateral PMTS is given by,
= [Area of Δ PMR] – [Area of Δ RST]
= 50 cm² – 8 cm²