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MATH / Three congruent isosceles triangles DAO, AOB, and OBC have AD=AO=OB=BC=10 and AB=DO=OC=12. These triangles are arranged to form trapezoid ABCD, as shown. Point P is on side AB so t…

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problem

Three congruent isosceles triangles DAO,DAO, AOB,AOB, and OBCOBC have AD=AO=OB=BC=10AD=AO=OB=BC=10 and AB=DO=OC=12.AB=DO=OC=12. These triangles are arranged to form trapezoid ABCD,ABCD, as shown. Point PP is on side ABAB so that OPOP is perpendicular to AB.AB. [asy] pair A, B, C, D, O, P; A= (6, 8); B=(18, 8); C=(24, 0); D=(0,0); O=(12,0); P=(12,8); draw(A--B--C--D--A); draw(A--O--B); draw(O--P, dashed); label("A", A, NW); label("B", B, NE); label("C", C, SE); label("D", D, SW); label("O", O, S); label("P", P, N); label("12", (D+O)/2, S); label("12", (O+C)/2, S); label("10", (A+D)/2, NW); label("10", (B+C)/2, NE); [/asy] What is the length of $OP?$
Plain-text mathematical notation (without MathML)
Three congruent isosceles triangles DAO, AOB, and OBC have AD=AO=OB=BC=10 and AB=DO=OC=12. These triangles are arranged to form trapezoid ABCD, as shown. Point P is on side AB so that OP is perpendicular to AB. [asy]
pair A, B, C, D, O, P;
A= (6, 8);
B=(18, 8);
C=(24, 0);
D=(0,0);
O=(12,0);
P=(12,8);
draw(A--B--C--D--A);
draw(A--O--B);
draw(O--P, dashed);
label("A", A, NW);
label("B", B, NE);
label("C", C, SE);
label("D", D, SW);
label("O", O, S);
label("P", P, N);
label("12", (D+O)/2, S);
label("12", (O+C)/2, S);
label("10", (A+D)/2, NW);
label("10", (B+C)/2, NE);
[/asy] What is the length of $OP?$
Original LaTeX notation
Three congruent isosceles triangles $DAO,$ $AOB,$ and $OBC$ have $AD=AO=OB=BC=10$ and $AB=DO=OC=12.$ These triangles are arranged to form trapezoid $ABCD,$ as shown. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB.$ [asy]
pair A, B, C, D, O, P;
A= (6, 8);
B=(18, 8);
C=(24, 0);
D=(0,0);
O=(12,0);
P=(12,8);
draw(A--B--C--D--A);
draw(A--O--B);
draw(O--P, dashed);
label("A", A, NW);
label("B", B, NE);
label("C", C, SE);
label("D", D, SW);
label("O", O, S);
label("P", P, N);
label("12", (D+O)/2, S);
label("12", (O+C)/2, S);
label("10", (A+D)/2, NW);
label("10", (B+C)/2, NE);
[/asy] What is the length of $OP?$

level

Level 2

type

Geometry

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Source and history

task_id
66e225c5-cfcd-5910-8fb6-d36a772aec92
task_key
geometry--test--66e225c5-cfcd-5910-8fb6-d36a772aec92
task_revision_id
1
config / split
geometry / test

Official source

initial import