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MATH / In the diagram, points X, Y and Z are on the sides of △UVW, as shown. Line segments UY, VZ and WX intersect at P. Point Y is on VW such that VY:YW=4:3. If △PYW has an area of 30 an…

Problem

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problem

In the diagram, points XX, YY and ZZ are on the sides of △UVW\triangle UVW, as shown. Line segments UYUY, VZVZ and WXWX intersect at PP. Point YY is on VWVW such that VY:YW=4:3VY:YW=4:3. If △PYW\triangle PYW has an area of 30 and △PZW\triangle PZW has an area of 35, determine the area of △UXP\triangle UXP. [asy] size(6cm); pair v = (0, 0); pair w = (10, 0); pair u = (3.5, 7); pair y = 4 * w / 7; pair x = 56 * u / 140; pair p = IP(w--x, u--y); pair z = IP(v--(10 * p), u--w); draw(u--v--w--cycle); draw(u--y);draw(x--w);draw(z--v); label("UU", u, N); label("XX", x, NW); label("PP", p, NE + 0.2 * W); label("ZZ", z, NE); label("VV", v, SW); label("YY", y, S); label("WW", w, SE);/* label("aa", centroid(p, v, y), fontsize(10)); label("bb", centroid(p, z, u), fontsize(10)); label("cc", centroid(p, u, x), fontsize(10)); label("dd", centroid(p, x, v), fontsize(10)); label("3030", centroid(p, y, w) + 0.2 * W, fontsize(10)); label("3535", centroid(p, z, w), fontsize(10));*/ [/asy]
Plain-text mathematical notation (without MathML)
In the diagram, points X, Y and Z are on the sides of △UVW, as shown.  Line segments UY, VZ and WX intersect at P.  Point Y is on VW such that VY:YW=4:3. If △PYW has an area of 30 and △PZW has an area of 35, determine the area of △UXP. [asy]
size(6cm);
pair v = (0, 0); pair w = (10, 0); pair u = (3.5, 7);
pair y = 4 * w / 7;
pair x = 56 * u / 140;
pair p = IP(w--x, u--y);
pair z = IP(v--(10 * p), u--w);
draw(u--v--w--cycle);
draw(u--y);draw(x--w);draw(z--v);

label("U", u, N);
label("X", x, NW);
label("P", p, NE + 0.2 * W);
label("Z", z, NE);
label("V", v, SW);
label("Y", y, S);
label("W", w, SE);/*
label("a", centroid(p, v, y), fontsize(10));
label("b", centroid(p, z, u), fontsize(10));
label("c", centroid(p, u, x), fontsize(10));
label("d", centroid(p, x, v), fontsize(10));
label("30", centroid(p, y, w) + 0.2 * W, fontsize(10));
label("35", centroid(p, z, w), fontsize(10));*/
[/asy]
Original LaTeX notation
In the diagram, points $X$, $Y$ and $Z$ are on the sides of $\triangle UVW$, as shown.  Line segments $UY$, $VZ$ and $WX$ intersect at $P$.  Point $Y$ is on $VW$ such that $VY:YW=4:3$. If $\triangle PYW$ has an area of 30 and $\triangle PZW$ has an area of 35, determine the area of $\triangle UXP$. [asy]
size(6cm);
pair v = (0, 0); pair w = (10, 0); pair u = (3.5, 7);
pair y = 4 * w / 7;
pair x = 56 * u / 140;
pair p = IP(w--x, u--y);
pair z = IP(v--(10 * p), u--w);
draw(u--v--w--cycle);
draw(u--y);draw(x--w);draw(z--v);

label("$U$", u, N);
label("$X$", x, NW);
label("$P$", p, NE + 0.2 * W);
label("$Z$", z, NE);
label("$V$", v, SW);
label("$Y$", y, S);
label("$W$", w, SE);/*
label("$a$", centroid(p, v, y), fontsize(10));
label("$b$", centroid(p, z, u), fontsize(10));
label("$c$", centroid(p, u, x), fontsize(10));
label("$d$", centroid(p, x, v), fontsize(10));
label("$30$", centroid(p, y, w) + 0.2 * W, fontsize(10));
label("$35$", centroid(p, z, w), fontsize(10));*/
[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
83620697-45dd-5f01-880f-2f1c8f464708
task_key
geometry--test--83620697-45dd-5f01-880f-2f1c8f464708
task_revision_id
1
config / split
geometry / test

Official source

initial import