1976 AMC 12 Problem 16

Question

In triangles

ABC

and

DEF,

lengths

AC,

BC,

DF

and

EF

are all equal. Length

AB

is twice the length of the altitude of

\triangle DEF

from

F

to

DE.

Which of the following statements is (are) true?

\mathrm{I}.

\angle ACB

and

\angle DFE

must be complementary.

\mathrm{II}.

\angle ACB

and

\angle DFE

must be supplementary.

\mathrm{III}.

The area of

\triangle ABC

must equal the area of

\triangle DEF.

\mathrm{IV}.

The area of

\triangle ABC

must equal twice the area of

\triangle DEF.

\mathrm{II}

only

\mathrm{III}

only

\mathrm{IV}

only

\mathrm{I}

and

\mathrm{III}

only

\mathrm{II}

and

\mathrm{III}

only

Answer: E

Concepts:isosceles triangle altitude congruence (geometry)area

Difficulty rating: 2130

Small hint:

Drop altitudes from

C

and

F

; each altitude bisects the base of its isosceles triangle

Big hint:

Compare a half of

ABC

with a half of

DEF

as right triangles

Solution:

Let the equal sides have length

s,

let

AB=2a,

and let the altitude from

F

be

a.

In the isosceles triangles, the altitude from

C

bisects

AB,

while the altitude from

F

bisects

DE.

A half of

ABC

has hypotenuse

s

and leg

a;

a half of

DEF

has the same hypotenuse and altitude leg

a.

The two right triangles are congruent with their legs interchanged. Thus their relevant acute angles are complementary, so the full vertex angles

\angle ACB

and

\angle DFE

sum to

180^\circ.

The corresponding half-areas are equal as well, hence the full triangle areas are equal. Statements

\mathrm{II}

and

\mathrm{III}

hold.

Therefore, the correct answer is E.

1976 AMC 12 Problem 16

Answer: E

Difficulty rating: 2130

Small hint:

Big hint:

Solution:

Problem 16 in other years