Circles and Geometric Shapes | Exercise 5.6

Question 3

Find xx in Fig. 5.26.

Question diagram 1
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Solution

We will use the property that opposite angles of a cyclic quadrilateral are supplementary.

Step 1 — Identify the shape

The points A, D, C, and B lie on the circle.

This means that ADCB is a cyclic quadrilateral.

Diagram 1

Step 2 — Apply the property

Opposite angles in a cyclic quadrilateral add up to 180∘180^\circ.

So, ∠ADC+∠ABC=180∘\angle ADC + \angle ABC = 180^\circ.

We are given ∠ADC=100∘\angle ADC = 100^\circ.

We need to find xx, which is ∠ABC\angle ABC.

100∘+x=180∘100^\circ + x = 180^\circ

x=180∘−100∘x = 180^\circ - 100^\circ

x=80∘\boxed{x = 80^\circ}

Answer

(i) x=80∘x = 80^\circ

More questions in Exercise 5.6

Q1

In a circle with centre O, the central angle AOB is 60°. If the radius of the circle is 12 cm, what is the length of the chord AB?

Q2

Let AA and BB be two points on a circle with centre OO.

(i) Are there points X,YX, Y on the circle, on the same side of ABAB, such that ∠AXB\angle AXB is different from ∠AYB\angle AYB?

(ii) Is it true that if ∠AXB=∠AYB\angle AXB = \angle AYB, then XX and YY lie on the same side of the circle?

(iii) If ∠AXB=∠AYB\angle AXB = \angle AYB, and XX and YY do not lie on the circle, does the circle through AA, BB and XX also pass through YY?

Q3

Find xx in Fig. 5.26.

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