**Chapter 1 Similarity Set 1.2**

## Chapter 1 Similarity Set 1.2

**Question 1.Given below are some triangles and lengths of line segments. Identify in which figures, ray PM is the bisector of âˆ QPR.**

Solution:

Solution:

**Question 2.In âˆ†PQR PM = 15, PQ = 25, PR = 20, NR = 8. State whether line NM is parallel to side RQ. Give reason.Solution:**

PN + NR = PR [P â€“ N â€“ R]

âˆ´ PN + 8 = 20

âˆ´ PN = 20 â€“ 8 = 12

Also, PM + MQ = PQ [P â€“ M â€“ Q]

âˆ´ 15 + MQ = 25

âˆ´ line NM || side RQ [Converse of basic proportionality theorem]

**Question 3.In âˆ†MNP, NQ is a bisector of âˆ N. If MN = 5, PN = 7, MQ = 2.5, then find QP.Solution:**

**Question 4.Solution:**

Proof

âˆ APQ = âˆ ABC = 60Â° [Given]

âˆ´ âˆ APQ â‰… âˆ ABC

âˆ´ side PQ || side BC (i) [Corresponding angles test]

In âˆ†ABC,

sidePQ || sideBC [From (i)]

**Question 5.In trapezium ABCD, side AB || side PQ || side DC, AP = 15, PD = 12, QC = 14, find BQ.Solution:**

side AB || side PQ || side DC [Given]

**Question 6.Find QP using given information in the figure.Solution:**

In âˆ†MNP, seg NQ bisects âˆ N. [Given]

**Question 7.In the adjoining figure, if AB || CD || FE, then find x and AE.Solution:**

line AB || line CD || line FE [Given]

âˆ´ X = 6 units

Now, AE AC + CE [A â€“ C â€“ E]

= 12 + x

= 12 + 6

= 18 units

âˆ´ x = 6 units and AE = 18 units

**Question 8.In âˆ†LMN, ray MT bisects âˆ LMN. If LM = 6, MN = 10, TN = 8, then find LT.Solution:**

In âˆ†LMN, ray MT bisects âˆ LMN. [Given]

**Question 9.In âˆ†ABC,seg BD bisects âˆ ABC. If AB = x,BC x+ 5, AD = x â€“ 2, DC = x + 2, then find the value of x.Solution:**

In âˆ†ABC, seg BD bisects âˆ ABC. [Given]

âˆ´ x(x + 2) = (x â€“ 2)(x + 5)

âˆ´ x2 + 2x = x2 + 5x â€“ 2x â€“ 10

âˆ´ 2x = 3x â€“ 10

âˆ´ 10 = 3x â€“ 2x

âˆ´ x = 10

**Question 10.In the adjoining figure, X is any point in the interior of triangle. Point X is joined to vertices of triangle. Seg PQ || seg DE, seg QR || seg EF. Fill in the blanks to prove that, seg PR || seg DF.Solution:**

**Question 11.In âˆ†ABC, ray BD bisects âˆ ABC and ray CE bisects âˆ ACB. If seg AB = seg AC, then prove that ED || BC.Solution:**

**Question 1.i. Draw a âˆ†ABC.ii. Bisect âˆ B and name the point of intersection of AC and the angle bisector as D.iii. Measure the sides.iv. Find ratios and v. You will find that both the ratios are almost equal.vi. Bisect remaining angles of the triangle and find the ratios as above. Verify that the ratios are equal. (Textbook pg. no. 8)Solution:**

Note: Students should bisect the remaining angles and verify that the ratios are equal.

**Question 2.Write another proof of the above theorem (property of an angle bisector of a triangle). Use the following properties and write the proof.i. The areas of two triangles of equal height are proportional to their bases.ii. Every point on the bisector of an angle is equidistant from the sides of the angle. (Textbook pg. no. 9)Given: In âˆ†CAB, ray AD bisects âˆ A.Construction: Draw seg DM âŠ¥ seg AB A â€“ M â€“ B and seg DN âŠ¥ seg AC, A â€“ N â€“ C.Solution:**

Proof:

In âˆ†ABC,

Point D is on angle bisector of âˆ A. [Given]

âˆ´DM = DN [Every point on the bisector of an angle is equidistant from the sides of the angle]

**Question 3.i. Draw three parallel lines.ii. Label them as l, m, n.iii. Draw transversals t**

_{1}and t

_{2}.iv. AB and BC are intercepts on transversal t

_{1}.v. PQ and QR are intercepts on transversal t

_{2}.vi. Find ratios AB/BC and PQ/QR. You will find that they are almost equal. Verify that they are equal.(Textbook pg, no. 10)Solution:

(Students should draw figures similar to the ones given and verify the properties.)

**Question 4.In the adjoining figure, AB || CD || EF. If AC = 5.4, CE = 9, BD = 7.5, then find DF. (Textbook pg, no. 12)Solution:**

**Question 5.In âˆ†ABC, ray BD bisects âˆ ABC. A â€“ D â€“ C, side DE || side BC, A â€“ E â€“ B, then prove that AB/BC = AE/EB (Textbook pg, no. 13)Solution:**