AMC 10 · 2024 · #15

Grade 11 geometry-2dalgebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Lattice or Non-Aligned Polygon Area

logarithm-propertiesshoelace-formulacoordinate-geometryarea-triangles convert-to-algebraidentify-subproblems ↑ Prerequisites: logarithm-propertiescoordinate-geometry

📏 Medium solution 💡 3 insights

Problem

A triangle in the coordinate plane has vertices A(log₂ 1, log₂ 2), B(log₂ 3, log₂ 4), and C(log₂ 7, log₂ 8). Find the area of △ ABC, written as a single base-2 logarithm.

Pick an answer.

(A)

$\log_2\frac{\sqrt3}7$

(B)

$\log_2\frac3{\sqrt7}$

(C)

$\log_2\frac7{\sqrt3}$

(D)

$\log_2\frac{11}{\sqrt7}$

(E)

$\log_2\frac{11}{\sqrt3}$

How to solve

Strategy Introduce a Variable

The problem looks like a logarithm problem, but underneath it is an ordinary coordinate-geometry problem wearing a disguise. Tool #15 (Organize Information in More Ways) strips most of the disguise: rewriting each label in simplest form turns four of the six coordinates into the integers 0, 1, 2, 3. Only log₂ 3 and log₂ 7 survive, and Tool #4 (Introduce a Variable) handles those by calling them p and q. They are fixed numbers, not unknowns to solve for, so naming them keeps the algebra readable. With A(0,1), B(p,2), C(q,3) the triangle is routine: Tool #1 (Draw a Diagram) shows it is a thin sliver, and Tool #13 (Convert to Algebra) applies the shoelace formula to get an exact expression in p and q. The last stretch is Tool #15 again, converting that expression back into the single-logarithm form the choices demand.

1STEP 1

Cash in the easy logarithms

The y-coordinates become 1, 2, 3.

A = (0, 1), B = (log₂ 3, 2), C = (log₂ 7, 3)

2STEP 2

Name the two leftover logarithms

Only the x-coordinates stay logarithmic.

p = log₂ 3, q = log₂ 7 → A(0,1), B(p,2), C(q,3)

3STEP 3

Plot the points and expect a sliver

The three points nearly form a straight line.

1 ＜ p ＜ 2, 2 ＜ q ＜ 3

4STEP 4

Shoelace from a corner at the origin

The area is half of two p minus q.

[ABC] = 1/2lvert p · 2 - q · 1 rvert = 1/2lvert 2p - q rvert

5STEP 5

Put the logarithms back and fix the sign

Confirm the value is positive.

1/2(2log₂ 3 - log₂ 7) = log₂ 3 - 1/2log₂ 7

6STEP 6

Fold it into one logarithm

It folds into log base 2 of 3 over root 7.

log₂ 3 - 1/2log₂ 7 = log₂ 3 - log₂ √(7) = log₂ 3/√(7)

Answer

log₂₃/√7

Check the closed form against the raw computation numerically. With p ≈ 1.5850 and q ≈ 2.8074, the shoelace value is 1/2lvert 2(1.5850) - 2.8074rvert ≈ 1/2(0.3626) ≈ 0.1813. Evaluating choice (B) straight from its own expression, 3/√(7) ≈ 1.1339 and log₂ 1.1339 ≈ 0.1813 — the two match, so the log arithmetic did not lose anything. The sketch predicted a thin sliver, and 0.1813 is indeed tiny. The other choices fail immediately: (A) is log₂ of a number smaller than 1, hence negative, and an area cannot be negative; (C), (D), (E) are log₂ of numbers between about 4 and 6.4, giving roughly 2.0 to 2.7, more than ten times too large. Note also that dropping the 1/2 would give log₂ 9/7, which appears nowhere in the list — the choices themselves flag that mistake.

💡Key takeaway

When coordinates are dressed up as logarithms, cash in the easy ones, give the stubborn ones short names, and do plain coordinate geometry — then use log rules at the end to turn your number back into the shape the choices are written in.

Cash in the easy logarithms
Name the two leftover logarithms
Plot the points and expect a sliver
Shoelace from a corner at the origin
Put the logarithms back and fix the sign
Fold it into one logarithm