Apply shared-base and shared-height reasoning, area ratios, side extensions, trapezoid and parallelogram networks, area ledgers, and multistep equal-area chains.
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Triangle ABC is divided by segment ED into Regions I and II. E lies between A and B, and D lies between B and C. Given
The area of Region II is ______ times the area of Region I.
Region I is triangle BED; Region II is the remaining part.
BD is a diagonal of trapezoid ABCD, whose parallel bases are AD and BC. E lies between B and C. Segment AE is parallel to leg DC, and AE intersects BD at O.
Find the area of trapezoid ABCD.
AE ∥ DC. Diagram not necessarily to scale.
In parallelogram ABCD, BC = 3 cm. The perpendicular height from A to BC is AE = 2 cm.
Find the area of triangle ACD.
AE ⟂ BC.
In parallelogram MNOP, Q is a point strictly between the endpoints of side OP. Lines MO and NQ intersect at R.
Fill each blank with >, <, or =.
Q lies strictly between O and P.
In parallelogram ABCD, E and F are the midpoints of AD and CD, respectively.
How many other triangles in the diagram have the same area as triangle BFC? Count triangles whose three sides follow drawn segments, including triangles made of several smaller pieces. Exclude the reference triangle itself.
E and F are midpoints. Count triangles other than △BFC itself.
In triangle ABC, D is the midpoint of BC, and point E lies on AD with
The area of triangle ABC is ______ times the area of triangle CDE.
D is the midpoint of BC.
In the diagram, points D, A, and C lie on one straight line in that order, while E lies on CB. Given
Complete the relationship: BE = ______ × BC.
The drawing shows the incidences, not the true scale.
Triangles ABC and BCD share base BC. Segments AD and BC intersect at E, with AE = ED and AD ⟂ BC.
Divide BC into eight equal parts. Let F be the first division point from B, and the intersection E is the second division point from B. Connect every division point to both A and D.
How many other triangles in the diagram have the same area as triangle ABF? Count triangles whose three sides follow drawn segments, including triangles made of several smaller pieces. Exclude the reference triangle itself.
F is the first eighth-point; E is the second.
In the trapezoid, the top base is parallel to BC and BC = 5. The perpendicular distance between the top base and BC is 5. The perpendicular distance from the intersection point A to BC is 2.
Find the sum of the areas of the two shaded triangles.
The two diagonals meet at A.
A large regular hexagon has area 24 cm². Three congruent small regular hexagons are placed inside it. Each small hexagon has two opposite vertices at the large hexagon’s center and at one of three alternating outer vertices, as shown. The small hexagons do not overlap.
Find the area of the shaded portion.
The three congruent small hexagons are shaded.
Give a final answer and show your reasoning.
In trapezoid ABCD, the upper base AD is 12 cm and the height BD is 18 cm. Diagonal AC intersects BD at E, and
Find the length of the lower base BC.
BD is perpendicular to BC.
In parallelogram ABCD, BC = 10 cm. Triangle ECB is right-angled at C, with EC = 8 cm. Line BE meets AD at F, and EC meets AD at G.
Find the area of parallelogram ABCD.
The shaded parts are △ABF and △GCD.
Triangle ABC has area 180 cm². Point D is the midpoint of BC. Point E lies on AD with
Point F lies on BE with EF = 3BF. Find the area of triangle AEF.
D is the midpoint of BC.
Square ABCD has side length 12. Point P is any point on AB.
Points M and N trisect BC; points I and H trisect AD. Points E, F, and G divide CD into four equal parts. The point orders are B–M–N–C, A–I–H–D, and C–E–F–G–D.
Find the total shaded area.
The shaded regions are quadrilateral DGPH and triangles FEP and MNP. P may be anywhere on AB.
F lies between A and D in rectangle ABCD. Segments BD and CF meet at E and divide the rectangle into four regions.
Find the area of quadrilateral ABEF (green).
The fourth region, triangle BEC, is unshaded.
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