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MATH / Let ABCD be a convex quadrilateral, and let M and N be the midpoints of (AC)¯ and (BD)¯, respectively. Then there exists a constant k so that AB²+BC²+CD²+DA²=AC²+BD²+k⋅MN².Find k. …

Problem

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

problem

Let ABCDABCD be a convex quadrilateral, and let MM and NN be the midpoints of AC¯\overline{AC} and BD¯,\overline{BD}, respectively. Then there exists a constant kk so that AB2+BC2+CD2+DA2=AC2+BD2+k⋅MN2.AB^2 + BC^2 + CD^2 + DA^2 = AC^2 + BD^2 + k \cdot MN^2.Find k.k. [asy] unitsize(0.8 cm); pair A, B, C, D, M, N; A = (0,0); B = (4,0.5); C = (5,-3); D = (-2,-2.5); M = (A + C)/2; N = (B + D)/2; draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(M--N); label("AA", A, NW); label("BB", B, NE); label("CC", C, SE); label("DD", D, SW); dot("MM", M, NE); dot("NN", N, NW); [/asy]
Plain-text mathematical notation (without MathML)
Let ABCD be a convex quadrilateral, and let M and N be the midpoints of (AC)¯ and (BD)¯, respectively.  Then there exists a constant k so that
AB²+BC²+CD²+DA²=AC²+BD²+k⋅MN².Find k.

[asy]
unitsize(0.8 cm);

pair A, B, C, D, M, N;

A = (0,0);
B = (4,0.5);
C = (5,-3);
D = (-2,-2.5);
M = (A + C)/2;
N = (B + D)/2;

draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
draw(M--N);

label("A", A, NW);
label("B", B, NE);
label("C", C, SE);
label("D", D, SW);
dot("M", M, NE);
dot("N", N, NW);
[/asy]
Original LaTeX notation
Let $ABCD$ be a convex quadrilateral, and let $M$ and $N$ be the midpoints of $\overline{AC}$ and $\overline{BD},$ respectively.  Then there exists a constant $k$ so that
\[AB^2 + BC^2 + CD^2 + DA^2 = AC^2 + BD^2 + k \cdot MN^2.\]Find $k.$

[asy]
unitsize(0.8 cm);

pair A, B, C, D, M, N;

A = (0,0);
B = (4,0.5);
C = (5,-3);
D = (-2,-2.5);
M = (A + C)/2;
N = (B + D)/2;

draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
draw(M--N);

label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, SE);
label("$D$", D, SW);
dot("$M$", M, NE);
dot("$N$", N, NW);
[/asy]

level

Level 3

type

Precalculus

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Source and history

task_id
57bba79d-fdb2-58be-9fac-642b0cf59682
task_key
precalculus--test--57bba79d-fdb2-58be-9fac-642b0cf59682
task_revision_id
1
config / split
precalculus / test

Official source

initial import