Examkin

Question

Difficulty: MediumQuadrilaterals and Polygons

In convex quadrilateral ABCDABCD, diagonal ACAC is drawn. It is given that AB=9AB = 9, BC=12BC = 12, CD=8CD = 8, and DA=15DA = 15, with ∠ABC=90∘\angle ABC = 90^\circ. Which of the following statements must be true? Select all that apply.

  1. The length of diagonal ACAC is 15.Answer
  2. B
    The area of quadrilateral ABCDABCD is 114.
  3. The perimeter of quadrilateral ABCDABCD is 44.Answer
  4. D
    A triangle can be formed using side lengths of 8, 15, and 32.
  5. The sum of the interior angles of quadrilateral ABCDABCD is 360∘360^\circ.Answer

Answer

The correct statements are that the length of diagonal ACAC is 15, the perimeter of quadrilateral ABCDABCD is 44, and the sum of the interior angles of quadrilateral ABCDABCD is 360∘360^\circ.
The statement specifying that the length of diagonal ACAC is 15 is correct because right triangle ABCABC has leg lengths 9 and 12, giving hypotenuse 81+144=15\sqrt{81 + 144} = 15. The statement specifying that the perimeter is 44 is correct because summing the outer side lengths yields 9+12+8+15=449 + 12 + 8 + 15 = 44. The statement specifying that the interior angle sum is 360∘360^\circ is correct because every convex quadrilateral has an interior angle sum of (4−2)×180∘=360∘(4 - 2) \times 180^\circ = 360^\circ.

Step-by-Step Solution

1
Calculate the length of diagonal ACAC using triangle ABCABC.
AC=AB2+BC2=92+122=15AC = \sqrt{AB^2 + BC^2} = \sqrt{9^2 + 12^2} = 15.
Since ∠ABC=90∘\angle ABC = 90^\circ, triangle ABCABC is a right triangle, allowing the application of the Pythagorean theorem.
2
Calculate the perimeter of quadrilateral ABCDABCD.
Perimeter =9+12+8+15=44= 9 + 12 + 8 + 15 = 44.
The perimeter of a polygon is the sum of all its outer side lengths.
3
Verify the sum of the interior angles for quadrilateral ABCDABCD.
Interior angle sum =(4−2)×180∘=360∘= (4 - 2) \times 180^\circ = 360^\circ.
The formula (n−2)×180∘(n - 2) \times 180^\circ applies to all convex polygons.
4
Evaluate the incorrect claims regarding area and triangle formation.
Triangle ADCADC is not a right triangle (82+152≠1528^2 + 15^2 \neq 15^2), so its area is not 60; and side lengths 8, 15, and 32 violate the triangle inequality theorem (8+15=23<328 + 15 = 23 < 32).
Right triangle formulas require a right angle, and valid triangle side lengths must satisfy the triangle inequality theorem.

Key Concept

Properties of convex quadrilaterals, right triangle side relationships, and triangle inequality bounds
Estimated Time:1m 30s
Rate this question

Topics

Quadrilaterals and Polygons

More questions from this topic