PQR is an isosceles triangle whose equal sides PQ and PR are at right angles. S and T are points on PQ such that $Q S = 6 S P$ and $Q T = 2 T P . P R S = θ, P R T = ϕ$

s i n 2 ϕ = 3 / 5

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2 ϕ + θ = 45^{o}

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s i n 2 θ = 3 / 5

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s i n ϕ + c o s ϕ = \frac{4}{\sqrt{10}}

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Solution

The correct options are
A $s i n 2 ϕ = 3 / 5$
B $2 ϕ + θ = 45^{o}$
D $s i n ϕ + c o s ϕ = \frac{4}{\sqrt{10}}$
Let equal sides be $a$
$tan θ = \frac{S P}{P R} = \frac{\frac{a}{2}}{a} = \frac{1}{2}$
$sin θ = \frac{1}{\sqrt{50}}$
$tan ϕ = \frac{P T}{P R} = \frac{\frac{a}{3}}{a} = \frac{1}{3}$
$sin ϕ = \frac{1}{\sqrt{10}}$
$sin 2 ϕ = \frac{2 tan ϕ}{1 + {tan}^{2} ϕ} = \frac{2 \times \frac{1}{3}}{1 + \frac{1}{9}} = \frac{3}{5}$
$cos 2 ϕ = \frac{4}{5}$
$sin ϕ + cos ϕ = \frac{1}{\sqrt{10}} + \frac{3}{\sqrt{10}} = \frac{4}{\sqrt{10}}$
$sin (2 ϕ + θ) = sin (2 ϕ) cos θ + cos 2 ϕ sin θ = \frac{3}{5} \times \frac{3}{\sqrt{50}} + \frac{4}{5} \times \frac{1}{\sqrt{50}} = \frac{1}{\sqrt{2}}$
Hence $2 ϕ + θ = {sin}^{- 1} \frac{1}{\sqrt{2}} = 45$

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Similar questions

S

and

T

are respectively the mid points of equal sides

P Q

and

P R

△ P Q R

. Show that

Q t = R S

Q. Let PQR be an isosceles triangles. Suppose that the sides PQ and PR are equal and let the length of PQ be K cm. If

∠ P Q R = θ, c o s θ = \frac{4}{5}

and area of triangle is M sq. cm, then which of the following is true about M?

Q. The sides PQ, PR of triangle PQR are equal, and S,T are points on PR,PQ such that

∠ P S Q a n d ∠ P T R

are right angles.
And hence, the triangles PSR and PSQ are congruent
If the above statement is true then mention answer as 1, else mention 0 if false

The sides PQ, PR of $Δ P Q R$ are equal, and S, T are points on PR, PQ such that $∠ P S Q$ and $∠ P T R$ are right angles. Hence, $Δ P T R ≅ Δ P S Q$

State whether the above statement is true or false.

Draw a triangle PQR such that PQ = 5 cm, P = $120^{\circ}$ and PR = 6 cm. Construct another triangle
Whose sides are $\frac{3}{4}$ times the corresponding sides of $Δ$ PQR?

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PQR is an isosceles triangle whose equal sides PQ and PR are at right angles. S and T are points on PQ such that QS=6SP and QT=2TP.PRS=θ,PRT=ϕ

PQR is an isosceles triangle whose equal sides PQ and PR are at right angles. S and T are points on PQ such that $Q S = 6 S P$ and $Q T = 2 T P . P R S = θ, P R T = ϕ$