Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

The centre of the circle passing through the midpoints of the sides...

The centre of the circle passing through the midpoints of the sides of an isosceles triangle

A B C

lies on the circumcircle of triangle

A B C

. If the larger angle of triangle

A B C

is

\alpha^0

and the smaller one

\beta^{\circ}

then what is the value of

\alpha-\beta

?

The centre of the circle passing through the midpoints of the sides of an isosceles triangle A B C lies on the

The centre of the circle passing through the midpoints of the sides of an isosceles triangle A B C lies on the

Correct answer

90

Step-by-step solution

The centre of the circle passing through the midpoints of the sides of an isosceles triangle A B C lies on the

\begin{aligned} & A+2 B=180 \\ & \frac{A}{2}=90-B \\ & 2 x+2 B=180 \\ & x=90-B \\ & x=\frac{A}{2} \\ & \Rightarrow O P \perp B C \end{aligned}

\therefore \mathrm{O}

and A coincide

\begin{aligned} & \therefore \angle \mathrm{ORP}=\angle \mathrm{OPR}=\angle \mathrm{RAP}=60^{\circ} \\ & \therefore \alpha=120^{\circ} \beta=30^{\circ} \\ & \alpha-\beta=90^{\circ} \end{aligned}

By drawing properly we get The centre of the circle passing through the midpoints of the sides of an isosceles triangle A B C lies on the

The centre of the circle passing through the midpoints of the sides of an isosceles triangle A B C lies on the

Practice Properties of Triangles on Quantrex Academy →

More from Properties of Triangles

The incircle Γ of a scalene triangle A B C touches B C at D, C A at E and A B at F . Let… Olympiad workbook The sides x and y of a scalene triangle satisfy x+(2 Δ)/(x)=y+(2 Δ)/(y) , where Δ is the… Olympiad workbook In a triangle A B C , the median A D divides ∠ B A C in the ratio 1: 2 . Extend A D to E… Olympiad workbook Let A B C be a triangle let D be a point on the segment B C such that A D=B C . Suppose ∠… Olympiad workbook Two sides of a regular polygon with n sides, when extended, meet at an angle of 28^ ∘ .… Olympiad workbook Let A B C be an isosceles triangle with A B=A C and incentre I . If A I=3 and the… Olympiad workbook Let A B C be an acute-angled triangle and P be a point in its interior. Let P_A, P_B… Olympiad workbook Let A B C be a triangle with ∠ B A C=90^ ∘ and D be the point on the side B C such that A… Olympiad workbook Full Properties of Triangles list All Olympiad workbook PYQs