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MATH / Isosceles right triangle ABC encloses a semicircle of area 2π. The circle has its center O on hypotenuse (AB)¯ and is tangent to sides (AC)¯ and (BC)¯. What is the area of triangle…

Problem

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

problem

Isosceles right triangle ABCABC encloses a semicircle of area 2π2\pi. The circle has its center OO on hypotenuse AB¯\overline{AB} and is tangent to sides AC¯\overline{AC} and BC¯\overline{BC}. What is the area of triangle ABCABC? [asy] pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(circle(o, 2)); clip(a--b--c--cycle); draw(a--b--c--cycle); dot(o); label("CC", c, NW); label("AA", a, NE); label("BB", b, SW); [/asy]
Plain-text mathematical notation (without MathML)
Isosceles right triangle ABC encloses a semicircle of area 2π. The circle has its center O on hypotenuse (AB)¯ and is tangent to sides (AC)¯ and (BC)¯. What is the area of triangle ABC? [asy]
pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2);
draw(circle(o, 2));
clip(a--b--c--cycle);
draw(a--b--c--cycle);
dot(o);
label("C", c, NW);
label("A", a, NE);
label("B", b, SW);
[/asy]
Original LaTeX notation
Isosceles right triangle $ABC$ encloses a semicircle of area $2\pi$. The circle has its center $O$ on hypotenuse $\overline{AB}$ and is tangent to sides $\overline{AC}$ and $\overline{BC}$. What is the area of triangle $ABC$? [asy]
pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2);
draw(circle(o, 2));
clip(a--b--c--cycle);
draw(a--b--c--cycle);
dot(o);
label("$C$", c, NW);
label("$A$", a, NE);
label("$B$", b, SW);
[/asy]

level

Level 3

type

Geometry

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Source and history

task_id
0d082a38-7cfb-510c-a010-6afead276a23
task_key
geometry--test--0d082a38-7cfb-510c-a010-6afead276a23
task_revision_id
1
config / split
geometry / test

Official source

initial import