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MATH / The area of triangle XYZ is 8 square inches. Points A and B are midpoints of congruent segments (XY)¯ and (XZ)¯. Altitude (XC)¯ bisects (YZ)¯. What is the area (in square inches) o…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

The area of triangle XYZXYZ is 8 square inches. Points AA and BB are midpoints of congruent segments XY¯\overline{XY} and XZ¯\overline{XZ}. Altitude XC¯\overline{XC} bisects YZ¯\overline{YZ}. What is the area (in square inches) of the shaded region? [asy] /* AMC8 2002 #20 Problem */ draw((0,0)--(10,0)--(5,4)--cycle); draw((2.5,2)--(7.5,2)); draw((5,4)--(5,0)); fill((0,0)--(2.5,2)--(5,2)--(5,0)--cycle, mediumgrey); label(scale(0.8)*"XX", (5,4), N); label(scale(0.8)*"YY", (0,0), W); label(scale(0.8)*"ZZ", (10,0), E); label(scale(0.8)*"AA", (2.5,2.2), W); label(scale(0.8)*"BB", (7.5,2.2), E); label(scale(0.8)*"CC", (5,0), S); fill((0,-.8)--(1,-.8)--(1,-.95)--cycle, white); [/asy]
Plain-text mathematical notation (without MathML)
The area of triangle XYZ is 8 square inches.  Points A and B are midpoints of congruent segments (XY)¯ and (XZ)¯.  Altitude (XC)¯ bisects (YZ)¯.  What is the area (in square inches) of the shaded region? [asy]
/* AMC8 2002 #20 Problem */
draw((0,0)--(10,0)--(5,4)--cycle);
draw((2.5,2)--(7.5,2));
draw((5,4)--(5,0));
fill((0,0)--(2.5,2)--(5,2)--(5,0)--cycle, mediumgrey);
label(scale(0.8)*"X", (5,4), N);
label(scale(0.8)*"Y", (0,0), W);
label(scale(0.8)*"Z", (10,0), E);
label(scale(0.8)*"A", (2.5,2.2), W);
label(scale(0.8)*"B", (7.5,2.2), E);
label(scale(0.8)*"C", (5,0), S);
fill((0,-.8)--(1,-.8)--(1,-.95)--cycle, white);
[/asy]
Original LaTeX notation
The area of triangle $XYZ$ is 8 square inches.  Points $A$ and $B$ are midpoints of congruent segments $\overline{XY}$ and $\overline{XZ}$.  Altitude $\overline{XC}$ bisects $\overline{YZ}$.  What is the area (in square inches) of the shaded region? [asy]
/* AMC8 2002 #20 Problem */
draw((0,0)--(10,0)--(5,4)--cycle);
draw((2.5,2)--(7.5,2));
draw((5,4)--(5,0));
fill((0,0)--(2.5,2)--(5,2)--(5,0)--cycle, mediumgrey);
label(scale(0.8)*"$X$", (5,4), N);
label(scale(0.8)*"$Y$", (0,0), W);
label(scale(0.8)*"$Z$", (10,0), E);
label(scale(0.8)*"$A$", (2.5,2.2), W);
label(scale(0.8)*"$B$", (7.5,2.2), E);
label(scale(0.8)*"$C$", (5,0), S);
fill((0,-.8)--(1,-.8)--(1,-.95)--cycle, white);
[/asy]

level

Level 2

type

Geometry

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Source and history

task_id
43d808f1-8e69-595e-8f81-4cf18ad0ec5c
task_key
geometry--test--43d808f1-8e69-595e-8f81-4cf18ad0ec5c
task_revision_id
1
config / split
geometry / test

Official source

initial import