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Online Quiz (Worksheet A B C D)

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High School Mathematics - 2
8.15 Areas of Similar Triangles

Theorem: The ratio of the areas of similar triangles is equal to the ratio of the squares of their corresponding sides.

Given: Triangle ABC and Triangle PQR are similar

To prove: (Area of triangle ABC)/(Area of triangle PQR) = AB2/PQ2 = BC2/QR2 = CA2/RP2
Construction: Draw AD perpendicular to BC and PS perpendicular to QR
Proof: (Area of triangle ABC)/(Area of triangle PQR) = (1/2 x BC x AD)/(1/2 x QR x PS)
or (area of ABC)/(area of PQR) = BC/QR x AD/PS ----------equation 1
Now in triangles ADB and PSQ
angle B = angle Q (they are similar triangles)
angle ADB = angle PSQ (90o)
Hence the triangles ADB and PSQ are similar
Consequently AD/PS = AB/PQ
But AB/BC = BC/QR ( As ABC and PQR are similar)
Hence AD/PS = BC/QR
---------equation 2
Substituting 2 in 1 we get
(Area of ABC)/(Area of triangle PQR) = BC/QR x BC/QR
(Area of ABC)/(Area of triangle PQR) = BC2/QR

Example: Prove that PM2 = QM x MR

Solution: As PM is perpendicular to QR we find that triangles PMQ and PMR are similar
Hence PM/MR = QM/PM (In similar triangles corresponding sides are proportional)
Hence PM2 = MR x QM

Directions: Solve the following problems. Also write at least 10 examples of your own.
Q 1: In triangle ABC, BE is perpendicular to AC, AD is perpendicular to BC. Show that the two triangles ADC and BEC are similar.

Q 2: ABD is a triangle in which angle DAB is 90 degrees. AC is perpendicular to DB. Prove that AB2 = BCxBD

Q 3: In triangle ABC, BE is perpendicular to AC, AD is perpendicular to BC. Show that CDxAB = CAx DE

Q 4: In triangle ABC, XY is ll AC. XY, divides the region into two equal areas. Determine AX/AB

Q 5: The areas of two similar triangles ABC and PQR are 64 cm2 and 121 cm2. If QR = 15.4 cm, find BC.
26.8 cm
29 cm
25 cm

Q 6: The perimeters of two triangles are 30cm and 20 cm. If one side of the first triangle is 12, determine the corresponding side of the second triangle.
10 cm
14 cm
8 cm

Q 7: ABCD is a trapezium in which AB ll CD. AB = 2CD. Find the ratio of the triangles of AOB and COD.

Q 8: A vertical stick 12 cm long casts a shadow 8cm long on the ground. at the same time a tower casts a shadow 40 cm on the ground. Determine the height of the tower.
60 cm
25 cm
15 cm

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Question 10: This question is available to subscribers only!

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