1) If two triangles are similar, the ratio of areas of these triangles

1.	1) If two triangles are similar, the ratio of areas of these triangles is equal to the ratioof the squares of their corresponding sides. – prove.
Answer» Answer: Triangle ABC ~ Triangle PQR(Given) Step-by-step explanation: To prove:- Ar. of tr. ABC/Ar. of tr. PQR = AB^2/PQ^2 = BC^2/QR^2 = CA^2/RP^2. Construction : 1) DRAW AD perpendicular to BC 2) Draw PM perpendicular to QR Note. Draw the constructions on two similar triangles Now PROOF : We know that, ar. of tr. = 1/2 × base × height. Therefore ar. of tr. ABC/ar. of tr. PQR = 1/2 BC × AD // 1/2 QR × PM = BC/QR × AD/PM. ...(i) Given, Tr. ABC ~ Tr. PQR Therefore, AB/PQ = BC/QR = CA/RP ...(ii) and Ang. B = Ang. Q ...(iii) In tr. ABD and tr. PQM, Ang. B = Ang. Q (from (iii)) and Ang. ADB = Ang. PMQ (each = 90°) Therefore, Tr. ABD ~ Tr. PQM (AA SIMILARITY criterion) Therefore, AD/PM = AB/PQ. ...(iv) From (i),(ii) and (iv), we get Ar. of tr. ABC/Ar. of tr. PQR = AB/PQ × AB/PQ = AB^2/PQ^2 ...(v) From (v) and (ii), Ar. of tr. ABC/Ar. of tr. PQR = AB^2/PQ^2 = BC^2/QR^2 = CA^2/RP^2 (PROVED) HOPE IT HELPS ! Thank you...

1) If two triangles are similar, the ratio of areas of these triangles is equal to the ratioof the squares of their corresponding sides. – prove.

Answer»

Answer:

Triangle ABC ~ Triangle PQR(Given)

Step-by-step explanation:

To prove:- Ar. of tr. ABC/Ar. of tr. PQR = AB^2/PQ^2 = BC^2/QR^2 = CA^2/RP^2.

Construction : 1) DRAW AD perpendicular to BC

2) Draw PM perpendicular to QR

Note. Draw the constructions on two similar triangles

Now PROOF : We know that, ar. of tr. = 1/2 × base × height.

Therefore ar. of tr. ABC/ar. of tr. PQR = 1/2 BC × AD // 1/2 QR × PM = BC/QR × AD/PM. ...(i)

Given,

Tr. ABC ~ Tr. PQR

Therefore, AB/PQ = BC/QR = CA/RP ...(ii)

and Ang. B = Ang. Q ...(iii)

In tr. ABD and tr. PQM,

Ang. B = Ang. Q (from (iii))

and Ang. ADB = Ang. PMQ (each = 90°)

Therefore, Tr. ABD ~ Tr. PQM (AA SIMILARITY criterion)

Therefore, AD/PM = AB/PQ. ...(iv)

From (i),(ii) and (iv), we get

Ar. of tr. ABC/Ar. of tr. PQR = AB/PQ × AB/PQ = AB^2/PQ^2

...(v)

From (v) and (ii),

Ar. of tr. ABC/Ar. of tr. PQR = AB^2/PQ^2 = BC^2/QR^2 = CA^2/RP^2 (PROVED)

HOPE IT HELPS !

Thank you...