Circle theorems with algebra

Circle theorems

(1MA1) · Higher

ORIGINAL

Link same-segment angles, a diameter, and a triangle angle sum in one algebraic chain.

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Question

Points

A

B

C

and

D

lie on a circle.

AB

is a diameter, and

A

and

D

lie in the same segment cut off by chord

BC

. Angle

BAC=(3x+4)^\circ

and angle

BDC=(5x-10)^\circ

. Find

x

and angle

ABC

Every step worked, with the reasoning.

1
$(3x+4)^\circ=(5x-10)^\circ$
Angles $BAC$ and $BDC$ subtend the same chord $BC$ in the same segment, so they are equal.
2
$14=2x$ , so $x=7$
Rearrange and solve the linear equation.
3
$\angle BAC=3(7)+4=25^\circ$
Substitute $x=7$ into either given angle.
4
$\angle ACB=90^\circ$
The angle subtended by diameter $AB$ at the circumference is a right angle.
5
$\angle ABC=180^\circ-90^\circ-25^\circ=65^\circ$
Angles in triangle $ABC$ sum to $180^\circ$ .

Answer: $x=7$ and $\angle ABC=65^\circ$ .

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