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MATH / Let ABCD be a parallelogram. We have that M is the midpoint of AB and N is the midpoint of BC. The segments DM and DN intersect AC at P and Q, respectively. If AC=15, what is QA? […

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problem

Let ABCDABCD be a parallelogram. We have that MM is the midpoint of ABAB and NN is the midpoint of BC.BC. The segments DMDM and DNDN intersect ACAC at PP and QQ, respectively. If AC=15,AC = 15, what is QAQA? [asy] pair A, B, C, D, pM, pN, O, P, Q; A = (25, 0) * dir(-20); B = (15, 0) * dir(60); C = (25, 0) * dir(160); D = (15, 0) * dir(-120); pM = 0.5 * A + 0.5 * B; pN = 0.5 * B + 0.5 * C; O = 0.25 * A + 0.25 * B + 0.25 * C + 0.25 * D; P = 0.33 * C + 0.67 * A; Q = 0.67 * C + 0.33 * A; draw(A--B--C--D--cycle); draw(A--C); draw(pM--D); draw(pN--D); label("AA", A, S); label("BB", B, NE); label("CC", C, NW); label("DD", D, S); label("MM", pM, NE); label("NN", pN, NW); label("PP", P, N); label("QQ", Q, NE); [/asy]
Plain-text mathematical notation (without MathML)
Let ABCD be a parallelogram. We have that M is the midpoint of AB and N is the midpoint of BC. The segments DM and DN intersect AC at P and Q, respectively. If AC=15, what is QA? [asy]
pair A, B, C, D, pM, pN, O, P, Q;
A = (25, 0) * dir(-20);
B = (15, 0) * dir(60);
C = (25, 0) * dir(160);
D = (15, 0) * dir(-120);
pM = 0.5 * A + 0.5 * B;
pN = 0.5 * B + 0.5 * C;
O = 0.25 * A + 0.25 * B + 0.25 * C + 0.25 * D;
P = 0.33 * C + 0.67 * A;
Q = 0.67 * C + 0.33 * A;
draw(A--B--C--D--cycle);
draw(A--C);
draw(pM--D);
draw(pN--D);
label("A", A, S);
label("B", B, NE);
label("C", C, NW);
label("D", D, S);
label("M", pM, NE);
label("N", pN, NW);
label("P", P, N);
label("Q", Q, NE);
[/asy]
Original LaTeX notation
Let $ABCD$ be a parallelogram. We have that $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC.$ The segments $DM$ and $DN$ intersect $AC$ at $P$ and $Q$, respectively. If $AC = 15,$ what is $QA$? [asy]
pair A, B, C, D, pM, pN, O, P, Q;
A = (25, 0) * dir(-20);
B = (15, 0) * dir(60);
C = (25, 0) * dir(160);
D = (15, 0) * dir(-120);
pM = 0.5 * A + 0.5 * B;
pN = 0.5 * B + 0.5 * C;
O = 0.25 * A + 0.25 * B + 0.25 * C + 0.25 * D;
P = 0.33 * C + 0.67 * A;
Q = 0.67 * C + 0.33 * A;
draw(A--B--C--D--cycle);
draw(A--C);
draw(pM--D);
draw(pN--D);
label("$A$", A, S);
label("$B$", B, NE);
label("$C$", C, NW);
label("$D$", D, S);
label("$M$", pM, NE);
label("$N$", pN, NW);
label("$P$", P, N);
label("$Q$", Q, NE);
[/asy]

level

Level 3

type

Geometry

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Source and history

task_id
8c9c1983-f03e-5142-8362-46e30fd41ddd
task_key
geometry--test--8c9c1983-f03e-5142-8362-46e30fd41ddd
task_revision_id
1
config / split
geometry / test

Official source

initial import