Khan/khan-exercises

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 Area of trapezoids, rhombi, and kites
randFromArray(metricUnits.concat([genericUnit]))
randRange(2, 8) randRange(2, 8) randRange(1, 6) randRangeNonZero(-2, 2) 1/2 * (B1 + B2) * H
What is the area of this figure?
init({ range: [[-4, max(B1, B2) + 4], [-1, H + 1]], scale: [30, 30] }); style({ stroke: BLUE, fill: "#eee"}); path([[0, 0], [B1, 0], [B2 + SH, H], [SH, H], [0, 0]]); label([B1/2, 0], B1 + "\\text{ " + UNIT + "}", "below"); label([B2/2 + SH, H], B2 + "\\text{ " + UNIT + "}", "above"); var x = min(B1, B2 + SH); line([x, 0], [x, H], { strokeDasharray: "." }); label([x, H/2], H + "\\text{ " + UNIT + "}", "right"); rightAngleBox([[0, 0], [x, 0]], [[x, 0], [x, H]], { stroke: GRAY, opacity: 0.5 }); parallel([[0, 0], [B1, 0]], 1); parallel([[SH, H], [B2 + SH, H]], 1);
K square plural_form(UNIT_TEXT)

This figure is a quadrilateral with a pair of parallel sides (the top and bottom sides), so it's a trapezoid.

area of a trapezoid = \dfrac12 \cdot (b_1 + b_2) \cdot h [Show me why]

Let's draw a line between the opposite ends of the two bases.

var showSubHint = function() { graph.subhint.show(); $("a[data-subhint='area-trapezoid']") .unbind("click", showSubHint) .click(hideSubHint); }; var hideSubHint = function() { graph.subhint.hide();$("a[data-subhint='area-trapezoid']") .unbind("click", hideSubHint) .click(showSubHint); }; graph.subhint = raphael.set().push( path([[0, 0], [B1, 0], [B2 + SH, H]], { stroke: BLUE, fill: ORANGE, opacity: 0.5 }), path([[SH, H], [B2 + SH, H], [0, 0]], { stroke: BLUE, fill: RED, opacity: 0.5 }) ); hideSubHint();

Notice that the line divides the trapezoid into two triangles: one triangle with base b_1 = B1, and another triangle with base b_2 = B2. Both triangles have height h = H.

The area of the trapezoid is equal to the sum of the areas of the two triangles.

A = \dfrac12 \cdot b_1 \cdot h + \dfrac12 \cdot b_2 \cdot h

Factor out \dfrac12 \cdot h to get the formula for the area of a trapezoid:

A = \dfrac12 \cdot h \cdot (b_1 + b_2) = \dfrac12 \cdot (b_1 + b_2) \cdot h

Now use this formula to calculate the trapezoid's area.

b_1 = B1

b_2 = B2

h = H

A = \dfrac12 \cdot (B1 + B2) \cdot H = K

randRange(1, 7) * 2 randRange(1, 7) * 2 randFromArray(["v", "h"]) rand(3) !== 0 ? randRange(1, 5) : D1/2 1/2 * D1 * D2 SH === D1/2
What is the area of this figure?
var range, v; var drawCongruencies, drawD1, drawD2; if (ORIENT === "h") { range = [[-1, D1 + 2], [-D2/2 - 1, D2/2 + 1]]; v = [[0, 0], [SH, D2/2], [D1, 0], [SH, -D2/2], [0, 0]]; drawCongruencies = function(style) { congruent([[0, 0], [SH, D2/2]], 1, style); congruent([[0, 0], [SH, -D2/2]], 1, style); congruent([[SH, D2/2], [D1, 0]], RHOMBUS ? 1 : 2, style); congruent([[SH, -D2/2], [D1, 0]], RHOMBUS ? 1 : 2, style); }; drawD1 = function(style) { return { label: label([D1/2, 0], D1 + "\\text{ " + UNIT + "}", style), path: path([[0, 0], [D1, 0]], style) }; }; drawD2 = function(style) { return { label: label([D1, 0], D2 + "\\text{ " + UNIT + "}", "right", style), path: path([[D1, -D2/2], [D1, D2/2]], style) }; }; } else { range = [[-D2/2 - 1, D2/2 + 1], [-1, D1 + 2]]; v = [[0, 0], [D2/2, SH], [0, D1], [-D2/2, SH], [0, 0]]; drawCongruencies = function(style) { congruent([[0, 0], [D2/2, SH]], 1); congruent([[0, 0], [-D2/2, SH]], 1); congruent([[D2/2, SH], [0, D1]], RHOMBUS ? 1 : 2); congruent([[0, D1], [-D2/2, SH]], RHOMBUS ? 1 : 2); }; drawD1 = function(style) { return { label: label([0, D1/2], D1 + "\\text{ " + UNIT + "}", style), path: path([[0, 0], [0, D1]], style) }; }; drawD2 = function(style) { return { label: label([0, D1], D2 + "\\text{ " + UNIT + "}", "above", style), path: path([[-D2/2, D1], [D2/2, D1]], style) }; }; } init({ range: range, scale: 20 }); path(v, { stroke: BLUE, fill: "#eee" }); drawCongruencies({ stroke: BLUE }); style({ stroke: BLUE, strokeDasharray: "." }, function() { graph.d1 = drawD1(); graph.d2 = drawD2(); }); rightAngleBox(graph.d1.path.graphiePath, graph.d2.path.graphiePath, { stroke: GRAY, opacity: 0.5 });
K square plural_form(UNIT_TEXT)

This figure is a quadrilateral with perpendicular diagonals and two pairs of congruent, adjacent sides, so it is a kite.

In fact, because this shape's sides are all congruent, it is also a rhombus.

area of a kite = \dfrac12 \cdot d_1 \cdot d_2 [Show me why]

The horizontal diagonal in the center splits the kite into two congruent triangles.

The vertical diagonal in the center splits the kite into two congruent triangles.

var showSubHint = function() { graph.subhint.show(); $("a[data-subhint='area-kite']") .unbind("click", showSubHint) .click(hideSubHint); }; var hideSubHint = function() { graph.subhint.hide();$("a[data-subhint='area-kite']") .unbind("click", hideSubHint) .click(showSubHint); }; if (ORIENT === "h") { graph.subhint = raphael.set().push( path([[0, 0], [SH, D2/2], [D1, 0], [0, 0]], { fill: ORANGE, opacity: 0.5 }), path([[0, 0], [SH, -D2/2], [D1, 0], [0, 0]], { fill: GREEN, opacity: 0.5 }) ); } else { graph.subhint = raphael.set().push( path([[0, 0], [D2/2, SH], [0, D1], [0, 0]], { fill: ORANGE, opacity: 0.5 }), path([[0, 0], [-D2/2, SH], [0, D1], [0, 0]], { fill: GREEN, opacity: 0.5 }) ); } hideSubHint();

Let d_1 = D1, the diagonal that bisects the kite. Then let d_2 = D2.

Notice that d_1 is the base of both triangles, and d_2 is the combined height of the two triangles, so \dfrac{d_2}{2} is the height of each triangle.

So the area of each triangle is:

A_T = \dfrac12 \cdot b \cdot h = \dfrac12 \cdot d_1 \cdot \dfrac{d_2}{2} = \dfrac14 \cdot d_1 \cdot d_2

The area of both triangles combined, 2A_T, is the total area of the kite:

2A_T = 2(\dfrac14 \cdot d_1 \cdot d_2) = \dfrac12 \cdot d_1 \cdot d_2 = A

Now use this formula to calculate the kite's area.

d_1 = D1

d_2 = D2

A = \dfrac12 \cdot D1 \cdot D2 = K

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