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Online Quiz (Worksheet A B C D)

Questions Per Quiz = 2 4 6 8 10

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 CDxAB = CAx DE
Answer:

Q 2: 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.
8 cm
14 cm
10 cm

Q 3: In triangle ABC, BE is perpendicular to AC, AD is perpendicular to BC. Show that CA x CE = CB x CD
Answer:

Q 4: The areas of two similar triangles ABC and PQR are 64 cm2 and 121 cm2. If QR = 15.4 cm, find BC.
29 cm
26.8 cm
25 cm

Q 5: In triangle ABC, XY is ll AC. XY, divides the region into two equal areas. Determine AX/AB
1/2
2/1
1/4

Q 6: ABD is a triangle in which angle DAB is 90 degrees. AC is perpendicular to DB. Prove that AB2 = BCxBD
Answer:

Q 7: 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.
25 cm
60 cm
15 cm

Q 8: In triangle ABC, BE is perpendicular to AC, AD is perpendicular to BC. Show that the two triangles ABC and DEC are similar
Answer:

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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