Math by First Principles
Level 3 · Challenge ← Back to lesson

Areas of Quadrilaterals: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered
Question 1 of 12
  1. 1

    A convex trapezoid has perpendicular height 66 cm and area 114114 square cm. Its longer parallel base exceeds its shorter base by 1010 cm. Find the shorter base.

    Parallel bases and perpendicular heightSchematic trapezoid, not to scale. Parallel top and bottom bases are labeled b cm and (b+10) cm. A dashed perpendicular between them is labeled 6 cm, with a right-angle square. b cm(b+10) cm6 cmNot to scale
  2. 2

    In a rhombus each side is 1010 cm and one full diagonal is 1212 cm. Find the area.

    Perpendicular diagonals partition the areaSchematic rhombus ABCD, not to scale. A is above O, B right, C below, D left. Diagonals AC and BD intersect at O with a right-angle square. BD is labeled 12 cm. AC is labeled d. A single tick on every side marks all four sides congruent.ABCDO12 cmdNot to scale
  3. 3

    In convex kite ABCDABCD, the equal-side pairs meet at AA and CC. Diagonals meet at OO. Given BOBO is 55 cm, AOAO is 44 cm, and area 7575 square cm, find OCOC.

    Perpendicular diagonals partition the areaSchematic convex kite ABCD, not to scale. A is above O, B right, C below, D left. Diagonals AC and BD intersect at O with a right-angle square. AO is 4 cm. Single side ticks pair AB with AD; double ticks pair BC with CD.ABCDO4 cmNot to scale
  4. 4

    In a convex quadrilateral, diagonal ACAC is 1212 cm and area 102102 square cm. The perpendicular distance from BB to ACAC is 55 cm. Find the perpendicular distance from DD to ACAC.

    One diagonal gives two trianglesSchematic convex quadrilateral ABCD, not to scale. AC is a horizontal diagonal labeled 12 cm. Vertex B lies above AC and vertex D below it. Dashed perpendicular heights from B and D to AC are labeled 5 cm and h. Right-angle squares mark their feet. BD is not drawn or asserted perpendicular to AC.ABCD12 cm5 cmhNot to scale
  5. 5

    A convex trapezoid has one leg perpendicular to the bases. Its bases are 99 cm and 1717 cm and its other leg is 1010 cm. Find its area.

    Parallel bases and perpendicular heightSchematic trapezoid, not to scale. Parallel top and bottom bases are labeled 9 cm and 17 cm. A dashed perpendicular between them is labeled h, with a right-angle square. The left leg is perpendicular to the bases. The right slanted side is labeled 10 cm.9 cm17 cmh10 cmNot to scale
  6. 6

    A rhombus has area 120120 square cm and side length 1313 cm. One diagonal is 1010 cm. What are the other diagonal and the height to a side?

    Perpendicular diagonals partition the areaSchematic rhombus ABCD, not to scale. A is above O, B right, C below, D left. Diagonals AC and BD intersect at O with a right-angle square. BD is labeled 10 cm. AC is labeled d. A single tick on every side marks all four sides congruent.ABCDO10 cmdNot to scale
  7. 7

    A convex trapezoid has area 9999 square cm. Its bases measure (x+2)(x+2) cm and (2x+1)(2x+1) cm and its height is 66 cm. Find both base lengths.

    Parallel bases and perpendicular heightSchematic trapezoid, not to scale. Parallel top and bottom bases are labeled (x+2) cm and (2x+1) cm. A dashed perpendicular between them is labeled 6 cm, with a right-angle square. (x+2) cm(2x+1) cm6 cmNot to scale
  8. 8

    A convex kite has area 126126 square cm. Its full diagonal ACAC is 2121 cm, and AC bisects BD at O. The point O divides AC into parts 66 cm and 1515 cm. Find BO.

    Perpendicular diagonals partition the areaSchematic convex kite ABCD, not to scale. A is above O, B right, C below, D left. Diagonals AC and BD intersect at O with a right-angle square. AC is labeled 21 cm. AO is 6 cm. OC is 15 cm. Single side ticks pair AB with AD; double ticks pair BC with CD.ABCDO21 cm6 cm15 cmNot to scale
  9. 9

    Which change to the givens guarantees that diagonal lengths pp and qq determine area 12pq\tfrac12pq for a convex quadrilateral?

  10. 10

    A convex quadrilateral has perpendicular diagonals. Their lengths are (x+1)(x+1) cm and 1212 cm. Its area is 5454 square cm. What is the diagonal length expressed by x+1x+1?

    Perpendicular diagonals partition the areaSchematic convex quadrilateral ABCD, not to scale. A is above O, B right, C below, D left. Diagonals AC and BD intersect at O with a right-angle square. BD is labeled 12 cm. AC is labeled (x+1) cm. No side equality or diagonal bisection is asserted.ABCDO12 cm(x+1) cmNot to scale
  11. 11

    A convex trapezoid has midsegment 1414 cm and area 112112 square cm. Its longer base is 1818 cm. Find its height and shorter base.

    Parallel bases and perpendicular heightSchematic trapezoid, not to scale. Parallel top and bottom bases are labeled b and 18 cm. A dashed perpendicular between them is labeled h, with a right-angle square. b18 cmhNot to scale
  12. 12

    In a convex quadrilateral, diagonal ACAC is 2020 cm. The perpendicular distances of BB and DD to AC are xx cm and (x+3)(x+3) cm. Its area is 130130 square cm. Find the two distances.

    One diagonal gives two trianglesSchematic convex quadrilateral ABCD, not to scale. AC is a horizontal diagonal labeled 20 cm. Vertex B lies above AC and vertex D below it. Dashed perpendicular heights from B and D to AC are labeled x cm and (x+3) cm. Right-angle squares mark their feet. BD is not drawn or asserted perpendicular to AC.ABCD20 cmx cm(x+3) cmNot to scale
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