Plane Geometry
- The three successive angles of a cyclic quadrilateral are in the ratio 1 : 3 : 4, find the measure of the fourth angle?
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As we know that ,
The sum of opposite angles of a concyclic quadrilateral is 180°.
Given , Ratio = 1 : 3 : 4
∠A : ∠B : ∠C = 1 : 3 : 4
∴ ∠A : ∠C = 1 : 4∠A = 1 × 180° = 36° 5
Correct Option: A
As we know that ,
The sum of opposite angles of a concyclic quadrilateral is 180°.
Given , Ratio = 1 : 3 : 4
∠A : ∠B : ∠C = 1 : 3 : 4
∴ ∠A : ∠C = 1 : 4∠A = 1 × 180° = 36° 5 ∠C = 4 × 180° = 144° 5
∴ ∠B = 3 × 36° = 108°
∴ ∠D = 180° – 108º = 72°
- ABCD is a cyclic quadrilateral. AB and DC when produced meet at P, if PA = 8 cm, PB = 6 cm, PC = 4 cm, then the length (in cm) of PD is
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As per the given in question , we draw a figure cyclic quadrilateral ABCD in which AB and DC when produced meet at point P ,
Given , PA = 8 cm, PB = 6 cm, PC = 4 cm
From figure
Clearly, AP × BP = PD × PC
⇒ 8 × 6 = PD × 4Correct Option: D
As per the given in question , we draw a figure cyclic quadrilateral ABCD in which AB and DC when produced meet at point P ,
Given , PA = 8 cm, PB = 6 cm, PC = 4 cm
From figure
Clearly, AP × BP = PD × PC
⇒ 8 × 6 = PD × 4⇒ PD = 8 × 6 = 12 cm. 4
- ABCD is a cyclic quadrilateral. Diagonals AC and BD meets at P. If ∠APB = 110° and ∠CBD = 30°, then ∠ADB measures
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According to question , we draw a figure of cyclic quadrilateral ABCD
∠APB = 110° = ∠CPD
∴ ∠APD = 180° – 110° = 70° = ∠ BPC
∴ ∠PCB = 180° – 70° – 30° = 80°Correct Option: D
According to question , we draw a figure of cyclic quadrilateral ABCD
∠APB = 110° = ∠CPD
∴ ∠APD = 180° – 110° = 70° = ∠ BPC
∴ ∠PCB = 180° – 70° – 30° = 80°
Angles subtended by same arcs at the circumference are equal.
∴ ∠ACB or ∠PCB = ∠ADB = 80°
- The point of intersection of the diagonals AC and BD of the cyclic quadrilateral ABCD is P. If ∠APB = 64° and ∠CBD = 28°, the measure of ∠ADB is
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On the basis of question we draw a figure of cyclic quadrilateral ABCD
Here , ∠APB = 64°
∠CBD = 28°
∠CBD = ∠CAD = 28°
∠APD = 180° – 64° = 116°Correct Option: B
On the basis of question we draw a figure of cyclic quadrilateral ABCD
Here , ∠APB = 64°
∠CBD = 28°
∠CBD = ∠CAD = 28°
∠APD = 180° – 64° = 116°
∴ ∠ADB = 180° – 116° – 28°
∠ADB = 180° – 144° = 36°
- ABCD is a cyclic quadrilateral and AD is a diameter. If ∠DAC = 55° then value of ∠ABC is
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According to question , we draw a figure of cyclic quadrilateral ABCD and AD is a diameter ,
In ∆ ACD
Given , ∠DAC = 55°
∠ACD = 90°
∠D = 180° – 55° – 90° = 35°Correct Option: C
According to question , we draw a figure of cyclic quadrilateral ABCD and AD is a diameter ,
In ∆ ACD
Given , ∠DAC = 55°
∠ACD = 90°
∠D = 180° – 55° – 90° = 35°
∴ ∠ABC + ∠ADC = 180°
⇒ ∠ABC = 180° – 35° = 145°
