benchmarks.wiki

benchmarks.wiki / Public workspace

MATH / In rectangle ABCD, AD=1, P is on (AB)¯, and (DB)¯ and (DP)¯ trisect ∠ADC. Write the perimeter of △BDP in simplest form as: w+(x⋅√(y))/(z), where w,x,y,z are nonnegative integers. W…

Problem

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

problem

In rectangle ABCDABCD, AD=1AD=1, PP is on AB¯\overline{AB}, and DB¯\overline{DB} and DP¯\overline{DP} trisect ∠ADC\angle ADC. Write the perimeter of △BDP\triangle BDP in simplest form as: w+x⋅yzw + \frac{x \cdot \sqrt{y}}{z}, where w,x,y,zw, x, y, z are nonnegative integers. What is w+x+y+zw + x + y + z? [asy] pair A,B,C,D,P; A=(0,5); B=(8,5); C=(8,0); D=(0,0); P=(3,5); draw(A--B--C--D--cycle); draw(P--D--B); label("AA",A,N); label("BB",B,N); label("CC",C,S); label("DD",D,S); label("PP",P,N); [/asy]
Plain-text mathematical notation (without MathML)
In rectangle ABCD, AD=1, P is on (AB)¯, and (DB)¯ and (DP)¯ trisect ∠ADC. Write the perimeter of △BDP in simplest form as:  w+(x⋅√(y))/(z), where w,x,y,z are nonnegative integers.  What is w+x+y+z?

[asy]
pair A,B,C,D,P;
A=(0,5);
B=(8,5);
C=(8,0);
D=(0,0);
P=(3,5);
draw(A--B--C--D--cycle);
draw(P--D--B);
label("A",A,N);
label("B",B,N);
label("C",C,S);
label("D",D,S);
label("P",P,N);
[/asy]
Original LaTeX notation
In rectangle $ABCD$, $AD=1$, $P$ is on $\overline{AB}$, and $\overline{DB}$ and $\overline{DP}$ trisect $\angle ADC$. Write the perimeter of $\triangle BDP$ in simplest form as:  $w + \frac{x \cdot \sqrt{y}}{z}$, where $w, x, y, z$ are nonnegative integers.  What is $w + x + y + z$?

[asy]
pair A,B,C,D,P;
A=(0,5);
B=(8,5);
C=(8,0);
D=(0,0);
P=(3,5);
draw(A--B--C--D--cycle);
draw(P--D--B);
label("$A$",A,N);
label("$B$",B,N);
label("$C$",C,S);
label("$D$",D,S);
label("$P$",P,N);
[/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
decf79fa-f016-5f0e-86b2-9a523b8b03dc
task_key
geometry--test--decf79fa-f016-5f0e-86b2-9a523b8b03dc
task_revision_id
1
config / split
geometry / test

Official source

initial import