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## Tamilnadu Samacheer Kalvi 10th Maths Solutions Chapter 4 Geometry Ex 4.1

Question 1.

Check whether the which triangles are similar and find the value of x.

Solution:

x = \(\frac{15}{6}\) = 2.5

Question 2.

A girl looks the reflection of the top of the lamp post on the mirror which is 6.6 m away from the foot of the lamppost. The girl whose height is 1.25 m is standing 2.5 m away from the mirror. Assuming the mirror is placed on the ground facing the sky and the girl, mirror and the lamppost are in a same line, find the height of the lamp post?

Solution:

In the picture ∆MLN, ∆MGRare similar triangles.

∴ Height of the lamp post is 3.3 m.

Question 3.

A vertical stick of length 6 m casts a shadow 400 cm long on the ground and at the same time a tower casts a shadow 28 m long. Using similarity, find the height of the tower.

Solution:

In the picture ∆ABC, ∆DEC are similar triangles.

Height of a tower = 42 m

Question 4.

Two triangles QPR and QSR, right angled at P and S respectively are drawn on the same base QR and on the same side of QR. If PR and SQ intersect at T, prove that PT × TR = ST × TQ.

Solution:

In ∆RPQ,

RP^{2} + PQ^{2} = QR^{2}

∴ PQ^{2} = QR^{2} – RP^{2} ………… (1)

In ∆TPQ,

TP^{2} + PQ^{2} = QT^{2}

∴ PQ2^{2} = QT^{2} – TP^{2} ………….. (2)

Equating (1) and (2) we get,

QR^{2} – RP^{2} = QT^{2} – TP^{2}

RP = RT + TP

∴ QR^{2} – (RT + TP)^{2} = QT^{2} – TP^{2}

∴ QR^{2} – RT^{2} – TP^{2} – 2RT.TP = QT^{2} – TP^{2}

QR^{2} = QT^{2} + RT^{2} + 2RT.TP …………. (5)

In ∆QSR,

QS^{2} + SR^{2} = QR^{2}

SR^{2} = QR^{2} – SR^{2} ………..(3)

In ∆TSR,

ST^{2} + SR^{2} = TR^{2}

∴ SR^{2} = TR^{2} – TS^{2} ………… (4)

Equating (3) and (4) we get

QR^{2} – SQ^{2} = TR^{2} – TS^{2}

SQ = QT + TS

∴ QR^{2} – (2T + TS)^{2} = TR^{2} – TS^{2}

QR^{2} – 2T^{2} – TS^{2} – 2QT.TS = TR^{2} – TS^{2}

∴ 2R^{2} = TR^{2} + QT^{2} + 2QT.TS ………… (6)

Now equating (5) – (6), we get

QT^{2} +RT^{2} + 2RT. TP = QT^{2} + RT^{2} + 2QT.TS

∴ PT.TR = ST.TQ

Hence proved.

Question 5.

In the adjacent figure, ∆ABC is right angled at C and DE⊥AB. Prove that ∆ABC ~ ∆ADE and hence find the lengths of AE and DE?

Solution:

In ∆ABC & ∆ADE

∠A is common & ∠C = ∠E = 90°

∴ by similarity

∆ABC ~ ∆ADE

Substituting the values of DE and AE in (1)

we can prove that

Question 6.

In the adjacent figure, ∆ACB ~ ∆APQ . If BC = 8 cm, PQ = 4 cm, BA = 6.5 cm and AP = 2.8 cm, find CA and AQ.

Solution:

∆ACB ~ ∆APQ

Question 7.

In figure OPRQ is a square and MLN = 90°. Prove that

(i) ∆LOP ~ ∆QMO

(ii) ∆LOP ~ ∆RPN

(iii) ∆QMO ~ ∆RPN

(iv) QR^{2} = MQ × RN.

Solution:

(i) In ∆LOP & ∆QMO, we have

∠OLP = ∠MQO (each equal to 90°)

and ∠LOP = ∠OMQ (corresponding angles)

∆LOP ~ ∆QMO (by AA criterion of similarity)

(ii) In ∆LOP & ∆PRN, we have

∠PLO = ∠NRP (each equal to 90°)

∠LPO = ∠PNR (corresponding angles)

∆LOP ~ ∆RPN

(iii) In ∆QMO & ∆RPN .

Since ∆LOP ~ ∆QMO and ∆LOP ~ ∆RPN

∠QMO ~ ∆RPN

(iv) We have

∆QMO ~ ∆RPN (using (iii))

\(\frac{\mathrm{MQ}}{\mathrm{RP}}=\frac{\mathrm{QO}}{\mathrm{RN}}\) (∵ PROQ is a square)

QR^{2} = MQ × RN. [RP = QO, QO = QR]

Question 8.

If ∆ABC ~ ∆DEF such that area of ∆ABC is 9 cm^{2} and the area of ∆DEF is 16 cm^{2} and BC = 2.1 cm. Find the length of EF.

Solution:

Since the area of two similar triangles is equal to the ratio of the squares of any two corresponding sides.

Question 9.

Two vertical poles of heights 6 m and 3 m are erected above a horizontal ground AC. Find the value of y.

Solution:

∆PAC, ∆QBC are similar 6 triangles

⇒ AC = AB + BC

= 2BC + BC

AC = 3BC

Substituting AC = 3BC in (1), we get

(AC)y = 6BC

3(BC)y = 6(BC)

y = \(\frac{6}{3}\) = 2m

Question 10.

Construct a triangle similar to a given triangle PQR with its sides equal to \(\frac{2}{3}\) of the corresponding sides of the triangle PQR (scale factor \(\frac{2}{3}\)).

Solution:

Given a triangle PQR, we are required to construct another triangle whose sides are \(\frac{3}{5}\) of the corresponding sides of the triangle PQR.

Steps of construction:

(1) Draw any ray QX making an acute angle with QR on the side opposite to the vertex P.

(2) Locate 3 (the greater of 2 and 3 in \(\frac{2}{3}\) ) points. Q_{1} Q_{2}, Q_{2} on QX so that QQ_{1} = Q_{1}Q_{2} = Q_{2}Q_{3}

(3) Join Q_{3}R and draw a line through Q_{2} (the second point, 2 being smaller of 2 and 3 in \(\frac{2}{3}\)) parallel to Q_{3}R to intersect QR at R’.

(4) Draw line through R’ parallel to the line RP to intersect QP at P’.

The ΔP’QR’ is the required triangle each of the whose sides is \(\frac{2}{3}\) of the corresponding sides of 3 ΔPQR.

Question 11.

Construct a triangle similar to a given triangle LMN with its sides equal to \(\frac{4}{5}\) of the corresponding sides of the triangle LMN

(scale factor \(\frac{4}{5}\)).

Solution:

Given a triangle LMN, we are required to construct another triangle whose sides are \(\frac{4}{5}\) of the corresponding sides of the ΔLMN.

Steps of construction:

(1) Draw any ray making an acute angle to the vertex L.

(2) Locate 5 points (greater of 4 and 5 in \(\frac{4}{5}\) ) M_{1}, M_{2}, M_{3}, M_{4}, and M_{5} and MX so that MM_{1} = M_{1}M_{2} = M_{2}M_{3} = M_{3}M_{4} = M_{4}M_{5}

(3) Join M_{5}N and draw a line parallel to M_{5}N through M_{4} (the fourth point, 4 being the smaller of 4 and 5 in \(\frac{4}{5}\)) to intersect MN atN’.

(4) Draw a line through N^{1} parallel to the line NL to intersect ML and L’. Then ΔL’MN’ is the required triangle each of the whose sides is \(\frac{4}{5}\) of the corresponding sides of ΔLMN.

Question 12.

Construct a triangle similar to a given triangle ABC with its sides equal to \(\frac{6}{5}\) of the corresponding sides of the triangle ABC

(scale factor \(\frac{6}{4}\) ).

Solution:

ΔABC is the given triangle. We are required to construct another triangle whose sides are \(\frac{6}{5}\) of the corresponding sides of the given triangle ABC

Steps of construction:

(1) Draw any ray BX making an acute angle with BC on the opposite side to the vertex A.

(2) Locate 6 points (the greater of 6 and 5 in \(\frac{6}{5}\) ) B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6} so that BB_{1} = B_{1}B_{2} = B_{2}B_{3} = B_{4}B_{5} = B_{5}B_{6}.

(3) Join B_{5} (the fifth point, 5 being smaller of 5 and 6 in \(\frac{6}{5}\)) to C and draw a live through B_{6} parallel to B_{5}C intersecting the extended line segment BC at C^{1}.

(4) Draw a line through C’ parallel to CA intersecting the extended line segment BA at A’.

Then ΔA’BC’ is the required triangle each of whose sides is \(\frac{6}{5}\) of the corresponding sides of the given triangle ABC.

Question 13.

Construct a triangle similar to a given triangle PQR with its sides equal to \(\frac{7}{3}\) of the corresponding sides of the triangle PQR (scale factor \(\frac{7}{3}\)).

Solution:

Given a triangle ΔPQR. We have to construct another triangle whose sides are \(\frac{7}{3}\) of the corresponding sides of the given ΔPQR.

Steps of construction:

(1) Draw any ray QX making an acute angle with QR on the opposite side to the vertex P.

(2) Locate 7 points (the greater of 7 and 3 in \(\frac{7}{3}\)) Q_{1}, Q_{2}, Q_{3}, Q_{4}, Q_{5}, Q_{6}, and Q_{7} so that QQ_{2} = Q_{1}Q_{2} = Q_{2}Q_{3} = Q_{3}Q_{4} = Q_{4}Q_{5} = Q_{5}Q_{6}

= Q_{6}Q_{7}

(3) Join Q_{3} to R and draw a line segment through Q_{7} parallel to Q_{3}R intersecting the extended line segment QR at R’.

(4) Draw a line segment through R’ parallel to PR intersecting the extended line segment QP at P’.

Then ΔP’QR’ is the required triangle each of whose sides is \(\frac{7}{3}\) of the corresponding sides of the given triangle.