12 multiple-choice questions, progressively harder.
D lies inside BC with BD to DC equal to 2 to 7. E is the midpoint of AD. If ABC has area 117 square cm, what is the area of BDE?
Solution
Correct answer: D
The common-height partition gives ABD two ninths of the total, or 26 square cm. Within ABD, BE joins B to midpoint E of AD, so it halves that area.
Two triangles have equal included angles. Their enclosing sides are 12 cm and x cm in the first, and 15 cm and 8 cm in the second. Their first-to-second area ratio is 7 to 10. What is x?
Correct answer: B
Use the products around the equal included angles.
The positive value is compatible with a nondegenerate triangle having the given angle.
Two triangles have the same included angle of 37 degrees and equal areas. Their enclosing side lengths are x cm and 18 cm in the first, and 12 cm and 15 cm in the second. Find x.
Correct answer: C
Equal included angles give the ratio of side products. Equal areas therefore give equal products.
D is inside BC, and E lies on AD with AE to ED equal to 2 to 7. ABC has area 135 square cm. What is the area of EBC?
Correct answer: A
Triangles BED and BAD share vertex B and have bases ED and AD on one line. Their area ratio is 7 to 9. Triangles CED and CAD have the same ratio with common vertex C.
Adding the two corresponding subareas gives the same ratio for EBC and ABC.
The three medians of ABC meet at G, with opposite side midpoints D, E and F. One small median region has area 4.25 square cm. What is the area inside ABC but outside quadrilateral AEGF?
AEGF contains two of the six small regions. The portion outside it contains the other four.
Two triangles have equal chosen bases and equal areas. Their heights, in cm, are 3x−13x-13x−1 and x+9x+9x+9. What is x?
Equal positive bases and equal areas force equal heights.
Both heights are then 14 cm.
Two triangles have equal included angles of 125 degrees. Their enclosing sides are 9 cm and b cm in the first, and 12 cm and 22 cm in the second. Their area ratio is 6 to 11. What is b?
The height argument also covers an obtuse included angle, using feet on the supporting line.
G is the centroid of ABC, whose area is 138 square cm. M is the midpoint of segment CG. What is the area of BGM?
Triangle BGC contains two median regions, so its area is one third of 138, or 46 square cm. Within BGC, BM is a median to side CG.
Two triangles have equal heights. Their first-to-second base ratio is 9 to 5, and the first area exceeds the second by 44 square cm. What are their areas, respectively?
The areas have ratio 9 to 5. The difference is four ratio parts, so one part is 44 divided by 4, or 11 square cm.
Two triangles have equal included angles. In first-to-second order, one pair of enclosing sides has ratio 5 to 8, and the other pair has ratio 7 to 3. The second triangle has area 72 square cm. What is the first area?
Multiply the two ordered side ratios because the included angles match.
Apply this ratio to the known second area.
D is inside BC, and BC is 24 cm. The areas of ABD and ADC are 42 and 70 square cm, respectively. Find BD and DC.
The subtriangles share the height from A to BC, so the base ratio is 42 to 70, or 3 to 5. Eight length parts total 24 cm, so one part is 3 cm.
Two triangles have the same included angle, whose measure is unspecified. The enclosing sides are 4 cm and 21 cm in the first, and 6 cm and 14 cm in the second. Which conclusion is determined?
The two enclosing side products are both 84, so the common-angle height factor cancels to give equal areas. An unspecified included angle still allows different perpendicular heights. The areas cannot be fixed at 42 square cm unless the included angle is right.
Reset this practice set?
This clears every answer you have given and starts the set again from question 1.