Question

in triangle pqr QR is bisected of angle pqr S lies on PR t is any point at qs such that the line through t is parallel to PQ and Intercept QR at k then triangle qkt is always
isoceles
equilateral
right angled
isoceles right angled​

Answer 1

In ΔPQR, we have

PQ=PR⇒∠R=∠Q

Now, ST∥QR.

⇒∠PST=∠PQR and ∠PTS=∠PRQ [∵ Corresponding angles are equal]

⇒∠PST=∠Q and ∠PTS=∠R

⇒∠PST=∠PTS

⇒PT=PS

Hence, triangle is right angled

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Answer 2

Let m be the slope of PQ, then

tan 45

o

=

∣

∣

∣

∣

∣

1+m(−2)

m−(−2)

∣

∣

∣

∣

∣

⇒1=

∣

∣

∣

∣

∣

1−2m

m+2

∣

∣

∣

∣

∣

⇒±1=

1−2m

m+2

⇒m=

3

1

or m=3

As PR also makes ∠45

o

with RQ,

∴ The above two values of m are for PQ and PR.

∴ Equation of PQ is as follows:

y−1=−

3

1

(x−2)

⇒x+3y−5=0

and equation of PR is:

⇒2x−y−5=0

∴ Combined equation of PQ and PR is

(x−3y−5)(3x−y−5)=0

⇒3x

2

−3y

2

+8xy−20x−10y+25=0