If in Fig 6.1, O is the point of intersection of two chords AB and CD such that OB = OD, then triangles OAC and ODB are
a. equilateral but not similar
b. isosceles but not similar
c. equilateral and similar
d. isosceles and similar

If in Fig 6.1, O is the point of intersection of two chords AB and CD such that OB = OD, then triangles OAC and ODB are

Solution:

Given, O is the point of intersection of two chords AB and CD.

Also, OB = OD

We have to find the type of triangles OAC and ODB.

Chord Intersection theorem states that when two chords inside the circle intersect each other then the product of their segments are equal.

So, the product of segments, OA × OB = OC × OD

So, OA = OC and OB = OD

In triangle OAC, the length of OA and OC are equal

So, the triangle is isosceles.

In triangle OBD, the length of OB and OD are equal

So, the triangle is isosceles.

By the property of similarity,

Similar triangles have congruent corresponding angles and the corresponding sides are in proportion.

From the given figure,

Vertically opposite angles are equal. So, ∠AOC = ∠BOD

Angles in the same segment are equal. So, ∠OAC = ∠ODB

OA = OB = OC = OD = radius of the circle

It is clear that the corresponding sides and the corresponding angles are equal.

So, the triangles OAC and OBD are similar.

Therefore, the triangles OAC and OBD are isosceles and similar.

✦ Try This: If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords

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

NCERT Exemplar Class 10 Maths Exercise 6.1 Sample Problem 1

If in Fig 6.1, O is the point of intersection of two chords AB and CD such that OB = OD, then triangles OAC and ODB are, a. equilateral but not similar, b. isosceles but not similar, c. equilateral and similar, d. isosceles and similar

Summary:

If in Fig 6.1, O is the point of intersection of two chords AB and CD such that OB = OD, then triangles OAC and ODB are isosceles and similar

☛ Related Questions:

If in Fig 6.1, O is the point of intersection of two chords AB and CD such that OB = OD, then triangles OAC and ODB are a. equilateral but not similar b. isosceles but not similar c. equilateral and similar d. isosceles and similar

If in Fig 6.1, O is the point of intersection of two chords AB and CD such that OB = OD, then triangles OAC and ODB are, a. equilateral but not similar, b. isosceles but not similar, c. equilateral and similar, d. isosceles and similar

If in Fig 6.1, O is the point of intersection of two chords AB and CD such that OB = OD, then triangles OAC and ODB are
a. equilateral but not similar
b. isosceles but not similar
c. equilateral and similar
d. isosceles and similar