In Fig. 6.19, PQR is a right triangle right angled at Q and QS ⊥ PR . If PQ = 6 cm and PS = 4 cm, find QS, RS and QR

Solution:

Given, PQR is a right triangle with Q at right angle.

Also, QS ⊥ PR

The length of the sides

PQ = 6 cm

PS = 4 cm

We have to find the length of QS, RS and QR.

In △PSQ,

PQ² = PS² + SQ²

(6)² = (4)² + SQ²

36 = 16 + SQ²

SQ² = 36 - 16

SQ² = 20

Taking square root,

SQ = 2√5 cm

We know that a perpendicular drawn from the vertex of the right angle of a right triangle to its hypotenuse divides the triangle into two triangles which are similar to the whole triangle and to each other.

So, SQ² = PS × SR

(2√5)² = 4 × SR

4(5) = 4SR

SR = 5 cm

In △QSR,

Given, QS⊥PR

So, ∠QSR = 90°

By pythagoras theorem,

QR² = SQ² + RS²

QR² = (2√5)² + (5)²

QR² = 4(5) + 25

QR² = 20 + 25

QR² = 45

Taking square root,

QR = 3√5 cm

Therefore, the length of QS, RS and QR are 2√5 cm, 5 cm and 3√5 cm.

✦ Try This: PQR is a right triangle right angled at Q and QS ⊥ PR . If PQ = 8 cm and PS = 6 cm, find QS, RS and QR.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 6

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NCERT Exemplar Class 10 Maths Exercise 6.4 Problem 10

In Fig. 6.19, PQR is a right triangle right angled at Q and QS ⊥ PR . If PQ = 6 cm and PS = 4 cm, find QS, RS and QR

Summary:

In Fig. 6.19, PQR is a right triangle right angled at Q and QS ⊥ PR . If PQ = 6 cm and PS = 4 cm, then the lengths of QS, RS and QR are 2√5 cm, 5 cm and 3√5 cm respectively

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