2017 Grade 9 Math Challenge – Elimination Round with answer key – Part II

Below are the 2017 Grade 9 Math Challenge – Elimination Round questions and answers – Part II. More reviewers can be found on the Past Tests and All Posts pages.

26.) Find all possible values of x in the proportion (x – 3) : (x – 1) = (x + 6) : 20.

27.) An angle in a quadrilateral has measures 60^\circ, while the others have degree measures in the ration 3 : 5 : 7. Find the measure of the largest angle.

28.) In rhombus PQRS, \angle QPR = 4 \angle QSR. Find \angle PQS.

29.) The diagonals of rhombus STAY intersect at X. If AX = m + n, YX = 12, SX = 4m - n, and TX = 4n, find the length (in units) of the shorter diagonal.

30.) Find the perimeter (in units) of the rhombus STAY in the previous problem.

31.) In parallelogram ABCD, AB = 9, BC = 4, CD = 4x - 3y, and DA = x + y. Find x.

32.) In parallelogram HIJK, \angle J = (5x + 30)^\circ, \angle k = (3x - 10)^\circ. Find x.

33.) (Figure 1) In parallelogram ABCD, \angle BAD = 130^\circ, \angle ABD = 20^\circ, and E is chosen on diagonal BD so that DE = DC. Find \angle CEB.

34.) In an iscosceles trapezoid, the lengths (in units) of the diagonals are 3x + 2 and 5x – 8. Find x.

35.) In trapezoid MNPQ where MN || PQ, MN = 4 and PQ = 9. If R and S are the midpoints of MQ and NP, respectively, find the length (in units) of RS.

36.) The diagonals of quadrilateral POST are perpendicular, and intersect at E. Suppose that TE = OE = 3, SE is twice as long as PE, and the area of the quadrilateral is 18 sq.units. Find the length (in units) of SE.

37.) Suppose $latex ST = 2, QR = 6, $ and $latex PT = 5. $ Find PR.

38.) Suppose $latex ST = 4, QR = 5, $ and the perimeter of \bigtriangleup PST is 14. Find the perimeter (in units) of \bigtriangleup PQR.

39.) Suppose PS = 4, SQ = 6, PT = x - 3, and TR = x. Find x.

40.) The sides of \bigtriangleup ABC are 9 cm, 10 cm, and 12 cm. If \bigtriangleup ABC ~ \bigtriangleup DEF, find the length of the longest side of \bigtriangleup DEF if its shortest side is 6 cm.

41.) Two similar triangles have lengths of corresponding sides in the ratio 4 : 5. Find the ratio of their areas.

42.) Given the points of A(0, 0), B(10, 0), C(15,0) and D(6, 6) on the plane, the point E is chosen in the first quadrant such that AD || BE and BD || CE. Find the coordinates of E.

43.) (Figure 2) In the figure, AB = AC and DB = DE. If \angle EDC = 20^\circ, \angle FAD = 30^\circ and \angle ABC = 50^\circ, which triangle (whose sides already drawn) is similar to \bigtriangleup ADF?

For problems 44 – 46

In \bigtriangleup ABC, \angle B = 90^\circ. Let E be the point on AC so that BE \bot AC

44.) Suppose AE = 9 and CE = 2. Find BE (in units).

45.) Suppose AE = 30 and CE = 6. Find BC (in units).

46.) Suppose AB = 12 and \frac{AE}{CE} = 8. Find CE.

47.) Two sides of a rectangle are 9 cm and 12 cm long. Find the length of a diagonal.

48.) A ladder is leaning against a vertical wall with the top 5 m above the ground. The top of the ladder slides all the way down the wall so that the bottom of the ladder slides 1 meter away fromthe wall. How long is the ladder?

49.) The two legs of a right triangle are in the ratio \frac{\sqrt{5}}{2}. If the hypotenuse is 9 units long, find the area (in square units) of the triangle.

50.) In \bigtriangleup ABC, \angle C = 90^\circ and cos A = \frac{3}{5}. Find cos B.

Answer key:

26.) x = 6, 9
27.) 140^\circ
28.) 18^\circ
29.) 10
30.) 52
31.) 3
32.) 20
33.) 105^\circ
34.) 5
35.) 6.5
36.) 4
37.) 15
38.) 17.5 or \frac{35}{2}
39.) 9
40.) 8 cm
41.) 16 : 25 or \frac{16}{25}
42.) (13, 3)
43.) \bigtriangleup BDC
44.) 3 \sqrt{2}
45.) 6 \sqrt{6}
46.) \sqrt{2}
47.) 15 cm
48.) 13 m
49.) 9 \sqrt{5}
50.) \dfrac{4}{5} or 0.8

View Part I here: 2017 Grade 9 Math Challenge – Elimination Round with answer key – Part I

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