12 multiple-choice questions, progressively harder.
Triangles ABC and DBC share base BC. A and D lie on a line parallel to BC. If triangle ABC has area 29.5 square cm, what is the area of triangle DBC?
Solution
Correct answer: B
The triangles share a base and have equal perpendicular distances to its line. Their areas are equal, so the second area is 29.5 square cm.
Two triangles have equal perpendicular heights. Their chosen base lengths are 6 cm and 17 cm. What is the first-to-second area ratio?
Correct answer: D
Cancel the common positive height and the factor one half. The area ratio is the ordered base ratio.
D is inside side BC of triangle ABC. BD is 7 cm and DC is 12 cm. If triangle ABD has area 42 square cm, what is the area of triangle ADC?
Correct answer: A
The two triangles share the perpendicular height from A to line BC. One ratio part is 42 divided by 7, or 6 square cm.
Two triangles have the same chosen base length. Their corresponding heights are 5 cm and 8 cm. What is the first-to-second area ratio?
Correct answer: C
For equal bases, the common factor one half times the base cancels. The areas are in the height ratio, in the requested order.
Triangle ABC has area 58 square cm, and AD is a median. What is the area of triangle ADC?
The midpoint D gives equal bases BD and DC with a common height. Each subtriangle is half the whole.
The three medians of triangle ABC meet at G. D, E and F are the opposite side midpoints. If triangle GBD has area 13 square cm, what is the area of ABC?
The three medians form six equal-area regions. GBD is one of them.
Two triangles have equal included angles. The enclosing sides are 3 cm and 14 cm in the first, and 7 cm and 9 cm in the second. What is their first-to-second area ratio?
Equal included angles give an area ratio from the enclosing side products.
Which extra information allows the products of two specified sides in each triangle to determine their area ratio?
Equal included angles make the perpendicular heights proportional to one pair of enclosing sides. The remaining pair then supplies the base ratio. The other conditions do not provide that height relationship.
If a and b are the first enclosing sides, and c and d are the second, the result is
D lies inside BC in triangle ABC. Why do triangles ABD and ADC have the same height when BD and DC are chosen as bases?
The bases BD and DC lie on the same supporting line, and both triangles have apex A. Their height is the perpendicular distance from A to that line, even if the foot lies outside one base segment.
Writing h for each corresponding height gives
Two triangles have equal positive heights and equal areas. What follows about their chosen base lengths?
Their area ratio equals their base ratio because the heights are equal. Equal areas give a ratio of one, so the bases have equal lengths.
The three medians of ABC meet at G. If ABC has area 126 square cm, what is the area of triangle BGC?
BGC consists of two of the six equal median regions. It contains one third of the total.
D lies on side BC of triangle ABC. Which given fact makes AD divide the triangle into two equal areas?
Equal lengths BD and DC give equal bases. Both triangles already have the common height from A to line BC, so the midpoint condition gives equal areas.
The needed base equality is
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