12 multiple-choice questions, progressively harder.
D lies inside BC with BD to DC equal to 3 to 5. Triangle ABC has area 104 square cm. What are the areas of ABD and ADC, respectively?
Solution
Correct answer: C
The eight base-ratio parts transfer to eight equal area parts. Each part is 104 divided by 8, or 13 square cm.
Two triangles have equal chosen base lengths. Their areas are 45 and 72 square cm, respectively. The first height is 10 cm. What is the second height?
Correct answer: A
The height ratio equals the area ratio because the bases match.
Two triangles have equal perpendicular heights and equal areas. Their bases, in cm, are 2x+32x+32x+3 and 5x−95x-95x−9. What is x?
Correct answer: D
Equal heights and equal areas force equal bases.
Both bases are then 11 cm, so they are positive.
Two triangles have equal included angles of 150 degrees. Their enclosing sides are 8 cm and 13 cm in the first, and 12 cm and 26 cm in the second. If the first area is 26 square cm, what is the second?
Correct answer: B
The obtuse angle still gives proportional perpendicular heights to the supporting base lines. Compare the products.
Therefore the second area is 3 times 26, or 78 square cm.
In the median partition of ABC, small triangle GBD has area 11.5 square cm. What are the area of quadrilateral AEGF and the area of ABC, respectively?
Quadrilateral AEGF consists of two small regions, while ABC consists of six.
Two triangles have equal positive heights. Their bases are x+2x+2x+2 and 2x−12x-12x−1, in cm, and their ordered area ratio is 3 to 4. What is x?
Transfer the area ratio to the bases, then cross-multiply.
The bases are then 7.5 cm and 10 cm, both positive.
D lies inside BC. The ratio BD to DC is 11 to 6, and ABC has area 68 square cm. The perpendicular foot from A lies beyond C. What are the areas of ABD and ADC, respectively?
Both heights reach the same supporting line BC, so an outside foot does not change the ratio. There are 17 parts, each of area 4 square cm.
ABC has area 156 square cm. AD is a median, and G is the centroid. What is the area of BGD?
Median AD gives triangle BAD half the total area, or 78 square cm. The ratio GD to AD is one to three. Triangles BGD and BAD use bases on line AD and share vertex B.
ABC has area 136 square cm. D is the midpoint of BC and F is the midpoint of AB. What is the area of AFD?
Median AD makes ABD half the whole, or 68 square cm. Inside ABD, DF joins D to midpoint F of AB, so it halves that area again.
Two triangles have equal heights and bases 4.5 cm and 7.5 cm. The smaller-base triangle has area 27 square cm. What is the other area?
The larger-to-smaller base ratio is 7.5 divided by 4.5, or five thirds. Multiply the known area by the same factor.
Two right triangles have enclosing perpendicular sides 6 cm and 22 cm in the first, and 8 cm and 11 cm in the second. What is the first-to-second area ratio?
Their equal included angles are right angles, so the two pairs of sides are base-height pairs. The factors one half cancel.
All three medians of ABC meet at G, with side midpoints D, E and F. Quadrilateral AEGF has area 38 square cm. What is the area of ABC?
AEGF occupies two of the six equal median regions, or one third of ABC.
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