Free Printable Training Test

1.

Let $\Delta ABC$ be a triangle with the medians from vertices $B$ and $C$ perpendicular.

Compute the value of the ratio $\frac{AB^2+AC^2}{BC^2}$ .

a)

2

b)

3

c)

5

d)

4

2.

Suppose that <ACD and <DCB have the same measure, the length of AC is 12, the length of BC is 5, the length of AB is 13. What is the length of

CD?

a)

$\frac{65}{17}$

b)

$\frac{60\sqrt[]{2}}{17}$

c)

$\frac{7\sqrt[]{5}}{2}$

d)

$\frac{7\sqrt[]{2}}{5}$

3.

In triangle ABD, BC is the angle bisector of <B. AB = 5, AC = 3, BC = 7. Find the length of CD.

a)

$\frac{147}{16}$

b)

$\frac{167}{15}$

c)

$\frac{25}{9}$

d)

$\frac{28}{5}$

4.

In triangle ABC, side AB = 6, side AC = 3, and side BC = 4. What is

the length of the angle bisector to side AB ?

a)

$\frac{3\sqrt[]{29}}{11}$

b)

$\frac{2\sqrt[]{37}}{9}$

c)

$\frac{2\sqrt[]{39}}{5}$

d)

$\frac{2\sqrt[]{39}}{7}$

5.

In triangle ABC, side AC and the perpendicular bisector of BC meet in point D, and BD bisects <ABC. If AD = 9 and DC = 7, what is the area of triangle ABD?

a)

$10\sqrt[]{14}$

b)

$14\sqrt[]{10}$

c)

$14\sqrt[]{15}$

d)

$14\sqrt[]{5}$

6.

In triangle ABC, AB = 6 units and AC = 3 units. Point D is on segment BC so that BD: DC = 2 : 1. If AD = 2 units, what is the shortest possible length of segment BC?

a)

$21\sqrt[]{7}$

b)

$3\sqrt[]{7}$

c)

$7\sqrt[]{3}$

d)

$4\sqrt[]{7}$

7.

In triangle PQR, PQ = 8 units and PR = 6 units. Point S is on segment QR so that QS: SR = 3 : 2. If PS = 4 units, what is the shortest possible length of segment QR?

a)

$5\sqrt[]{7}$

b)

$2\sqrt[]{14}$

c)

$4\sqrt[]{6}$

d)

$6\sqrt[]{5}$

8.

In triangle XYZ, side XY = 10, side XZ = 4, and side YZ = 7. What is the length of the angle bisector to side XY?

a)

$\frac{4\sqrt[]{21}}{7}$

b)

$\frac{3\sqrt[]{31}}{8}$

c)

$\frac{2\sqrt[]{41}}{9}$

d)

$\frac{5\sqrt[]{19}}{6}$

9.

Let $\Delta MNO$ be a triangle with the medians from vertices $M$ and $O$ perpendicular. Compute the value of the ratio $\frac{MN^2+MO^2}{NO^2}$ .

a)

2

b)

3

c)

4

d)

5

10.

In $\Delta ABC$ , angle B is trisected by BD and BE which meet AC at D and E respectively. If BA = 4, BD = 3, BE = 5, BC = 6, what is the value of $\frac{AD}{CE}$ ?

a)

$\frac{1}{4}$

b)

$\frac{1}{2}$

c)

$\frac{3}{7}$

d)

$\frac{2}{5}$

11.

In triangle DEF, DE = 8 units and DF = 6 units. Point G is on segment EF so that EG: GF = 4 : 3. If DG = 5 units, what is the shortest possible length of segment EF?

a)

$3\sqrt[]{13}$

b)

$2\sqrt[]{15}$

c)

$5\sqrt[]{8}$

d)

$4\sqrt[]{10}$

12.

In triangle GHI, GH = 7 units and GI = 5 units. Point J is on segment HI so that HJ: JI = 5 : 2. If GJ = 3 units, what is the shortest possible length of segment HI?

a)

$2\sqrt[]{21}$

b)

$3\sqrt[]{22}$

c)

$\sqrt[]{52}$

d)

$4\sqrt[]{14}$

13.

In triangle KLM, KL = 9 units and KM = 7 units. Point N is on segment LM so that LN: NM = 3 : 4. If KN = 6 units, what is the shortest possible length of segment LM?

a)

$5\sqrt[]{12}$

b)

$2\sqrt[]{34}$

c)

$3\sqrt[]{21}$

d)

$\sqrt[]{75}$

14.

In $\Delta$ ABC, the ratio AC : CB = 4 : 5. The bisector of the exterior angle at C intersects BA extended at P (A is between P and B). The ratio PA : PB is:

a)

$\frac{2}{5}$

b)

$\frac{4}{5}$

c)

$\frac{1}{2}$

d)

$\frac{2}{3}$

15.

In triangle PQR, if the angle bisector of <PQR and <PRQ meet at point S inside the triangle, and PS = 8, SR = 6, and SQ = 7, what is the length of PR?

a)

$10$

b)

$12$

c)

$14$

d)

$16$

16.

In triangle XYZ, side XZ is extended to point W, making ZW twice as long as XZ. If XY = 10 units, YZ = 6 units, and XZ = 4 units, what is the length of segment WY?

a)

$8$

b)

$10$

c)

$12$

d)

$14$

17.

In triangle ABC, the angle bisector of <BAC intersects BC at point D. If AB = 7, AC = 5, and BC = 8, what is the length of AD?

a)

$\frac{56}{13}$

b)

$\frac{65}{14}$

c)

$\frac{70}{15}$

d)

$\frac{75}{16}$

18.

In right triangle ABC , M and N are midpoints of legs AB and BC ,

respectively. Leg AB is 6 units long, and leg BC is 8 units long. How many square units are in the area of $\Delta$ APC ?

a)

7

b)

8

c)

6

d)

18

19.

In triangle ABC, if the angle bisector of <BAC and <BCA meet at point D inside the triangle, and AD = 10, BD = 8, and CD = 6, what is the length of BC?

a)

$14$

b)

$12$

c)

$16$

d)

$18$

20.

In triangle XYZ, if XY = 7 units and XZ = 9 units. Point W is on segment YZ so that YW: WZ = 4 : 5. If XW = 8 units, what is the shortest possible length of segment YZ?

a)

$5\sqrt[]{13}$

b)

$2\sqrt[]{22}$

c)

$3\sqrt[]{17}$

d)

$4\sqrt[]{14}$

21.

In triangle PQR, the perpendicular bisectors of PQ and PR intersect at point S inside the triangle. If PQ = 12 units, PR = 10 units, and PS = 8 units, what is the area of triangle PQR?

a)

$48$

b)

$60$

c)

$72$

d)

$84$

Training Test

Total questions: 21

Worksheet time: 2hrs 39mins

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