AMC 10 · 2021 · #8

Grade 11 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

sine-area-formulaisosceles-trianglearea-trianglesangle-sum-triangle identify-subproblemsconvert-to-algebra ↑ Prerequisites: sine-area-formulaisosceles-triangle

📏 Medium solution 💡 3 insights

Problem

An isosceles triangle has two equal sides and a base, and the height to the base is the perpendicular distance from the apex down to it. This triangle is obtuse. Multiplying the two equal sides together gives the same number as multiplying the base by double the height. Find the angle at the apex.

Pick an answer.

(A)

105

(B)

120

(C)

135

(D)

150

(E)

165

How to solve

Strategy Organize Information in More Ways

The given condition mixes two things that do not normally sit in the same equation: a product of two sides on one side, and a base-times-height product on the other. Tool #15 (Organize Information in More Ways) is what unlocks it — the phrase "the base times twice the height" is not really about lengths, it is the area of the triangle wearing a disguise, since area is 1/2 times base times height. Once the condition is read as a statement about area, the same area can be measured a second way, from the two congruent sides and the angle between them, and the two measurements can be set equal. Tool #4 (Introduce a Variable) supplies the names needed to write that down and then lets the side length divide out, which it must, because the problem supplies no numeric lengths. Tool #3 (Eliminate Possibilities) finishes: the resulting equation has two candidate angles, and the single word "obtuse" is there to delete one of them.

1STEP 1

Name the four measurements

Name the four measurements.

s · s = b · 2h, that is s² = 2bh

2STEP 2

Read 2bh as four areas

Read the base-height product as an area.

Area = 1/2bh → bh = 2 Area → 2bh = 4 Area, so s² = 4 Area

3STEP 3

Measure the same area from the apex

Measure that area from the apex.

Area = 1/2 · s · s · sinθ = 1/2s²sinθ

4STEP 4

Divide the size away

Dividing makes the size vanish.

s² = 4 · 1/2s²sinθ = 2s²sinθ → 1 = 2sinθ → sinθ = 1/2

5STEP 5

Find every angle with sine one half

Two angles have that sine.

sin 30° = 1/2 and sin 150° = sin(180° - 30°) = 1/2 → θ = 30° or θ = 150°

6STEP 6

Let "obtuse" pick the winner

Being obtuse picks 150 degrees.

θ = 30° → (30°, 75°, 75°) — all acute, rejected; θ = 150° → (150°, 15°, 15°) — obtuse, accepted → (D)

Answer

150

Build the triangle and check it directly. Take s = 1 with vertex angle 150°; the base angles are 15° each. Splitting the triangle down its height gives b = 2sin 75° ≈ 1.9319 and h = cos 75° ≈ 0.2588, so b · 2h ≈ 1.9319 · 0.5176 ≈ 1.0000, which matches s² = 1. The condition holds. The other choices can be ruled out in one line each, since the condition reduces to 2sinθ = 1: at 105°, 120°, 135°, 165° the value of 2sinθ is about 1.932, 1.732, 1.414, 0.518 — none of them 1. Exactly one choice survives, which is what a well-posed multiple-choice problem should do. One more guard worth noting: the problem says the triangle is obtuse but never says the vertex angle is the obtuse one. That does not matter here, because the rejected root 30° produces a 30-75-75 triangle with no obtuse angle anywhere, so it fails under either reading.

💡Key takeaway

When a problem multiplies a base by a height, it is really talking about area — rewrite it that way, then measure the same area a second time from a different pair of sides, and setting the two measurements equal does the rest of the work.

Name the four measurements
Read 2bh as four areas
Measure the same area from the apex
Divide the size away
Find every angle with sine one half
Let "obtuse" pick the winner