In the figure (i) given below, PQ = 24 cm, QR = 7 cm and ∠PQR = 90°. Find the radius of the inscribed circle of △PQR.

Let the sides of triangle PQ, QR and PR meet the circle at L, M and N respectively. In right-angled triangle PQR From figure, RM = RN = (∵ tangents drawn from a common external point to a circle are equal.) RM = RQ - QM = (7 - x) cm. PL = PN = (∵ tangents drawn from a common external point to a circle are equal.) PL = PQ - QL = (24 - x) cm. We can see, PR = PN + RN = PL + RM. ⇒ 25 = 24 - x + 7 - x ⇒ 25 = 31 - 2x ⇒ 2x = 31 - 25 ⇒ 2x = 6 ⇒ x = 3. Hence, the radius of the inscribed circle is 3 cm.

In the figure (i) given below, the sides of the quadrilateral touch the circle. Prove that AB + CD = BC + DA.

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In the figure (ii) given below, two concentric circles with centre O are of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find BP.

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In the figure (i) given below, AB = 8 cm and M is mid-point of AB. Semicircles are drawn on AB, AM and MB as diameters. A circle with centre C touches all three semicircles as shown, find its radius.

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In the figure (ii) given below, ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6 cm (not drawn to scale). Three circles are drawn touching each other with vertices A, B and C as their centres. Find the radii of the three circles.

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Mathematics

In the figure (i) given below, PQ = 24 cm, QR = 7 cm and ∠PQR = 90°. Find the radius of the inscribed circle of △PQR.

Circles

Related Questions

In the figure (i) given below, the sides of the quadrilateral touch the circle. Prove that AB + CD = BC + DA.

In the figure (ii) given below, two concentric circles with centre O are of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find BP.

In the figure (i) given below, AB = 8 cm and M is mid-point of AB. Semicircles are drawn on AB, AM and MB as diameters. A circle with centre C touches all three semicircles as shown, find its radius.

In the figure (ii) given below, ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6 cm (not drawn to scale). Three circles are drawn touching each other with vertices A, B and C as their centres. Find the radii of the three circles.