Question

ans the question please​

Answer 1

Theorem related to triangle

Theorem1) The ratio of perimeter of two similar triangle is equal to the ratio of their corresponding side

Theorem 2) the ratio of area of two similar triangles is equal to the ratio of their square of corresponding side

Theorem 3) ratio of area of two similar triangles is equal to the ratio of square of perimeter of triangle.

Let us first prove theorem1)

we know that ratio of corresponding sides of similar triangle are equal

so,

AB/PQ =BC/QR=AC/PR

if we take out the ratio of the ratio of their corresponding sides we get

AB/PQ :BC/QR :AC/PR

=1:1:1

so, from ratio proportion theorem we can say

(AB+BC+AC)/(PQ+QR+PR) =AB/PQ=BC/QR=AC/PR

so,first theorem is proved.....

Let prove the second theorem

Consider the ∆ABC and ∆PQR again

draw AX and PY perpendicular to BC and QR respectively

now in. ∆ABX and in ∆PQY, we can see

angleABX=anglePQY (∆ABC and ∆PQR are similar)

also,angleAXB=anglePYQ (both are eqaul to 90)

by AA similarity we can say

both the traingle are similar

so,

AB/PQ=AX/PY

also,AB/PQ=BC/QR=AC/PR

so,

AX/PY=AB/PQ=BC/QR=AC/PR

ar(∆ABC)=1/2×BC×AX

ar(∆PQR)=1/2×QR×PY

taking the ratio of both the area we can say

ar(∆ABC)/are(∆PQR)=BC/QR ×AX/PY

also we know that

BC/QR=AX/PY

hence

ar(∆ABC)/are(∆PQR)=BC/QR ×BC/QR=(BC/QR)^2

hence second theorem is proved

If we relate the first and second theorem we can say,,,third theorem is exactly right

From,the theorem3 (which is proved above)

we can say

$\frac{ABC}{PQR} = \frac{ {(35)}^{2} }{ {(45)}^{2} } \\ = > \frac{ABC}{PQR} = \frac{49}{81}$

This will be the required ratio of area of triangle

Answer 2

Answer:

Since ∆ABC and ∆PQR are similar

the ratio of there corresponding sides is same

AB/PQ = BC/QR = AC/PR

Let

AB/PQ = BC/QR = AC/PR = K

∴AB = KPQ

BC = KQR

AC = KPR

Now perimeter of ∆ABC = AB + BC + AC

= KPQ + KQR + KPR

= K (PQ+QR+PR)

= K × Perimeter of ∆ PQR

Therefore,

Perimeter of ∆ABC/Perimeter of ∆PQR = K

So we can write,

AB/PQ = BC/QR = AC/PR = K. ...(1)

We know that

Ratio of area of similar triangles is equal to square of its corresponding sides.

Therefore,

Area of ∆ABC/Area of ∆PQR = (AB/PQ)²

Hence by putting the value of AB and PQ is 35cm and 45cm

Area of ∆ABC/Area of ∆PQR = (35/45)²

= (7/9)²

= 49/81

So, the area of ratio of two similar triangles is 49:81.

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