benchmarks.wiki

benchmarks.wiki / Public workspace

MATH / In the diagram, two circles, each with center D, have radii of 1 and 2. The total area of the shaded region is (5)/(12) of the area of the larger circle. How many degrees are in th…

Problem

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

problem

In the diagram, two circles, each with center DD, have radii of 11 and 22. The total area of the shaded region is 512\frac5{12} of the area of the larger circle. How many degrees are in the measure of (the smaller) ∠ADC\angle ADC? [asy] size(4cm); defaultpen(linewidth(0.75)); real adc = 100; pair d = (0, 0); pair a = 2 * dir(100); pair c = (2, 0); path inner = arc(d, a/2, c/2, CW); path outer = arc(d, c, a, CCW); guide region1 = (a--a/2)..inner..(c/2--c)..outer..cycle; guide region2 = arc(d, a/2, c/2, CCW)..(c/2--d--a/2)..cycle; fill(region1, gray(0.75)); fill(region2, gray(0.75)); draw(unitcircle); draw(scale(2) * unitcircle); draw(a--d--c); label("AA", a, N); label("CC", c, E); label("DD", d, NE); [/asy]
Plain-text mathematical notation (without MathML)
In the diagram, two circles, each with center D, have radii of 1 and 2. The total area of the shaded region is (5)/(12) of the area of the larger circle. How many degrees are in the measure of (the smaller) ∠ADC?
[asy]
size(4cm);
defaultpen(linewidth(0.75));
real adc = 100;
pair d = (0, 0); pair a = 2 * dir(100); pair c = (2, 0);
path inner = arc(d, a/2, c/2, CW);

path outer = arc(d, c, a, CCW);
guide region1 = (a--a/2)..inner..(c/2--c)..outer..cycle;
guide region2 = arc(d, a/2, c/2, CCW)..(c/2--d--a/2)..cycle;
fill(region1, gray(0.75));
fill(region2, gray(0.75));
draw(unitcircle); draw(scale(2) * unitcircle);
draw(a--d--c);
label("A", a, N); label("C", c, E); label("D", d, NE);
[/asy]
Original LaTeX notation
In the diagram, two circles, each with center $D$, have radii of $1$ and $2$. The total area of the shaded region is $\frac5{12}$ of the area of the larger circle. How many degrees are in the measure of (the smaller) $\angle ADC$?
[asy]
size(4cm);
defaultpen(linewidth(0.75));
real adc = 100;
pair d = (0, 0); pair a = 2 * dir(100); pair c = (2, 0);
path inner = arc(d, a/2, c/2, CW);

path outer = arc(d, c, a, CCW);
guide region1 = (a--a/2)..inner..(c/2--c)..outer..cycle;
guide region2 = arc(d, a/2, c/2, CCW)..(c/2--d--a/2)..cycle;
fill(region1, gray(0.75));
fill(region2, gray(0.75));
draw(unitcircle); draw(scale(2) * unitcircle);
draw(a--d--c);
label("$A$", a, N); label("$C$", c, E); label("$D$", d, NE);
[/asy]

level

Level 4

type

Geometry

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
2cfade28-0c92-5e8d-aba6-095fa557c649
task_key
geometry--test--2cfade28-0c92-5e8d-aba6-095fa557c649
task_revision_id
1
config / split
geometry / test

Official source

initial import