# 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.

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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