Angle in a semicircle is a right angle Archives – Mathemafia

Class 9 | Chapter 10 | Circles | Exercise 10.5 | Question 11

Sunil Syal — Fri, 04 Sep 2020 14:15:01 +0000

Question 11: ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.

This question is not correct!!!

While visualising the different scenarios of this question, I realised that there is something wrong in this question. This question is not correct for all possible scenarios, hence NCERT team need to fix this question.

There are three possible scenarios on the basis of the statement. There is one scenario out of three possible scenarios for which this statement is not valid.

It is request to the readers to share this question so that it reaches NCERT team and necessary changes are made.

Prerequisite

You must be aware of the following theorems to solve this question:

Same segment angles are always equal
Angle in a semicircle is a right angle

Theorem 10.9: Angles in the same segment of a circle are equal.

Angle in a semicircle is a right angle

Chapter 10 – Circles

Class 9 | Chapter 10 | Circles | Exercise 10.5 | Question 10

Sunil Syal — Fri, 04 Sep 2020 14:14:11 +0000

Question 10: If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Prerequisite

It is bit tricky question to solve (not difficult though). Such questions are generally proved by assuming the other way and proving the assumption wrong. For example, in this question we assumed that the point of intersection does not lie on the third side of the triangle and then proved this assumption wrong. To prove the theorem, we used one very common property of circles – angle in a semicircle is always a right angle.

Angle in a semicircle is a right angle

Chapter 10 – Circles

Theorems

Exercise 10.1

Question 1 & 2

Exercise 10.2

Question 1

Question 2

Exercise 10.3

Question 1

Question 2

Question 3

Exercise 10.4

Exercise 10.5

Examples

The post Class 9 | Chapter 10 | Circles | Exercise 10.5 | Question 10 appeared first on Mathemafia.

Class 9 | Chapter 10 | Circles | Angle in a semicircle is a right angle

Sunil Syal — Sun, 23 Aug 2020 09:03:12 +0000

Theorem: Prove that angle in a semicircle is a right angle. OR

Prove that angle subtended by a diameter is always a right angle.

The proof of this theorem 10.8 is a prerequisite to prove this theorem, in which we proved that the angle subtended at center of a circle by any arc is double the angle subtended anywhere on rest of the circle. Using that result we can easily prove this theorem.