Prerequisite Path · 8 Building Blocks + 1 Target Problem + 3 Bonus Problems
Three Equilateral Triangles, One Hidden Area — Solving for 11/12
Even a student brand-new to geometry can climb from angles → properties of equilateral triangles → the coordinate plane → area ratios, one step at a time,
until they can solve — on their own — a real problem built from three triangles and a single midpoint. Climb the staircase below, one card at a time.
8Building Blocks
1Target Problem
3Bonus Problems
1
Unit 1 · Basics of Angles
Straight Angles and Angle Addition
ConceptAn angle formed along a straight line is called a straight angle, and it always measures 180°. If two angles sit side by side along one straight line, their measures always add up to 180°.
Worked Example (shown before you try)Example. Along a straight line, one angle is 130°. What is the other angle? Solution. 180° − 130° = 50°. (A straight angle is always 180°, so subtract the known angle from it.)
Two angles along a straight line add up to 180°. If one angle measures 72°, what is the other angle?
180° − 72° = 108°. This "straight angle = 180°" rule will also anchor our reasoning when we look at the three angles of an equilateral triangle.
2
Unit 2 · Angle Sum of a Triangle
Add Up All Three Angles
ConceptThe three interior angles of any triangle always add up to 180°, no matter its shape.
Worked ExampleExample. Two angles are 55° and 65°. What is the third? Solution. 180° − 55° − 65° = 60°
Two interior angles of a triangle are 70° and 55°. What is the third angle?
180° − 70° − 55° = 55°
3
Unit 3 · Properties of Equilateral Triangles
Equal Area Means Equal Sides
ConceptAn equilateral triangle has three equal sides and three 60° angles. Here's a crucial property: if two equilateral triangles have the same area, their side lengths must also be equal (the triangles are fully congruent).
Worked ExampleExample. One side of an equilateral triangle is 5 cm. What are the other two sides, and what are the three angles? Solution. The other two sides are also 5 cm, and all three angles are 60°.
Equilateral triangle P and equilateral triangle Q have equal areas. What can we say about their side lengths?
Equal-area equilateral triangles must have equal side lengths. This is exactly the key that unlocks today's target problem: it tells us triangles ADE, CDE, and BCE all share the same side length.
4
Unit 4 · The Coordinate Plane
Describing a Point's Position With Numbers
ConceptA point's position on the coordinate plane is written as an ordered pair (x, y). x is the left-right position, y is the up-down position.
Worked ExampleExample. Where is the point (3, 0)? Solution. It's 3 units to the right of the origin.
What are the coordinates of the point exactly halfway between A(−2,0) and B(2,0)?
The average of the two x-coordinates −2 and 2 is 0, so the midpoint is (0,0).
5
Unit 5 · The Midpoint Formula
Midpoint = Average of Coordinates
ConceptFor two points (x₁,y₁) and (x₂,y₂), the midpoint is ( (x₁+x₂)/2 , (y₁+y₂)/2 )
Worked ExampleExample. What is the midpoint of A(0,0) and B(6,0)? Solution. ((0+6)/2, (0+0)/2) = (3,0)
What is the midpoint of A(1,4) and B(5,8)?
((1+5)/2, (4+8)/2) = (3,6). We'll use this exact formula later to find the coordinates of H, the midpoint of BC.
6
Unit 6 · Area of a Triangle
Base × Height ÷ 2
ConceptArea of a triangle = base × height ÷ 2
Worked ExampleExample. A triangle has base 6 and height 4. What is its area? Solution. 6 × 4 ÷ 2 = 12
A triangle has base 10 and height 3. What is its area?
10 × 3 ÷ 2 = 15
7
Unit 7 · Area Ratio of Triangles With Equal Height
Base Ratio = Area Ratio
ConceptIf two triangles have the same height, the ratio of their areas is exactly equal to the ratio of their bases.
Worked ExampleExample. Two triangles share the same height, and their bases are in ratio 2:3. What is the ratio of their areas? Solution. The area ratio is also 2:3.
A point on a segment splits it into a ratio of 3:1. Using that segment as the base, two triangles share the opposite vertex (so they have equal height). What is the ratio of their areas?
Since the base splits 3:1, the areas split in the same ratio: 3:1.
8
Unit 8 · The Key Theorem
Area Ratio of Triangles Sharing One Angle
ConceptIf two triangles share one vertex angle, and the two sides forming that angle are in ratios a:b and c:d, then the ratio of their areas is (a×c) : (b×d).
Worked ExampleExample. Two triangles share angle E. One pair of sides is in ratio 1:2, the other pair is in ratio 1:3. What is the area ratio? Solution. Area ratio = (1×1):(2×3) = 1:6
Two triangles share angle P. One pair of sides is in ratio 1:3, the other pair is in ratio 1:4. What is the ratio of their areas?
(1×1) : (3×4) = 1 : 12. This exact theorem is the final key to unlocking today's target problem.
★
Target Problem · Today's Challenge
Triangles ADE, CDE, BCE — What Is the Blue Area?
Triangles ADE, CDE, and BCE are all equilateral triangles with area 1, and H is the midpoint of BC. Segment AH is drawn. What is the area of the blue region — the part of triangle CDE that lies above segment AH — shown shaded in the figure?
[Solution] Since all three triangles have area 1 (Unit 3), all three side lengths are equal too. Let that side length be 1, and place E at the origin (Units 4–5):
A(−1,0), B(1,0), D(−1/2, √3/2), C(1/2, √3/2), and since H is the midpoint of BC, H(3/4, √3/4).
Let Y be where line AH crosses side DE, and let X be where line AH crosses side CE. Solving the resulting equations gives exactly
EY : ED = 1 : 4 and EX : EC = 1 : 3.
By the theorem from Unit 8, since triangles EXY and EDC share angle E, their area ratio is
(1/4)×(1/3) = 112. Since triangle EDC has area 1, triangle EXY has area 112.
So the blue region (quadrilateral DCXY) = 1 − 1/12 = 1112
+1
Bonus 1 · The Sliver That Got Cut Off
What Is the Area of Triangle EXY?
In the target problem above, segment AH slices off a small triangle EXY near vertex E of triangle CDE. What is the area of that small triangle? (Triangle CDE has area 1.)
Since EY:ED = 1:4 and EX:EC = 1:3, the area ratio is (1/4)×(1/3) = 112
+2
Bonus 2 · Applying the Theorem With New Numbers
Same Theorem, Different Triangle
In triangle PQR, point M lies on side PQ with PM:MQ = 1:2. Point N lies on side PR with PN:NR = 1:1. What is the ratio of the area of triangle PMN to the area of triangle PQR?
PM:PQ = 1:3 and PN:PR = 1:2, so the area ratio is (1×1):(3×2) = 1:6
+3
Bonus 3 · Solving It All Over Again With New Numbers
What If Each Area Were 3?
Suppose equilateral triangles ADE, CDE, and BCE each have area 3 instead of 1 (everything else is exactly the same as the target problem — H is still the midpoint of BC). What is the area of the blue quadrilateral now?
The ratio (11/12) stays the same no matter how big the triangle is. 3 × 11/12 = 114
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