Question

in the triangle PQR, angle PQR = 90 degrees,s, and RPQ = 31. The length of PQ is 11 cm.
Calculate:
a) the length of QR
b) the length of PR
c) the length of the perpendicular from Q to PR.

Answer 1

Answer: where is questions..... Please try again

Step-by-step explanation:

Answer 2

Answer:

The length of QR, PR and the perpendicular from Q to PR are 6.61cm, 12.83cm and 12.83cm respectively.

Step-by-step explanation:

Given in the triangle PQR,

Angle PQR = 90° and angle RPQ = 31°

The length of PQ is 11 cm.

Sum of the angles in a triangle is 180°.

Therefore

$\angle PQR + \angle RPQ +\angle QRP= 180^o$

$\implies 90^o+ 31^o +\angle QRP= 180^o$

$\implies\angle QRP=180^o-121^o=59^o$

Applying the sine rule, which gives the relation between the lengths of the sides of a triangle and the sines of its angles.

According to the sine rule,

$\dfrac{PR}{sin Q} =\dfrac{PQ}{sin R} = \dfrac{QR}{sin P}$

$\implies \dfrac{PR}{sin 90}=\dfrac{11}{sin 59} = \dfrac{QR}{sin 31}$

a) From

$\dfrac{11}{sin 59} = \dfrac{QR}{sin 31}\implies {11}\times{sin 31}= {QR}\times{sin 59}$

$\implies {11}\times0.515= {QR}\times0.857$

$\implies {QR}=\dfrac{{11}\times0.515}{0.857} =\dfrac{5.665}{0.857}=6.61cm$

Therefore, the length of QR is 6.61cm.

b) From

$\implies \dfrac{PR}{sin 90}=\dfrac{11}{sin 59}$

$\implies \dfrac{PR}{1}=\dfrac{11}{0.857}$

$\implies PR=\dfrac{11}{0.857}=12.83cm$

Therefore, the length of PR is 12.83cm.

c) Let us drop a perpendicular from S on PR to Q, then triangle SPQ and triangle PQR are similar, and hence

$\dfrac{QS}{QP}=\dfrac{QR}{RP}\implies QS=\dfrac{QR}{RP}\times QP$

$\implies QS=\dfrac{6.61}{12.83}\times 11=\dfrac{72.71}{12.83}\approx5.66cm$

Therefore, the length of the perpendicular from Q to PR is 12.83cm.

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