Question

Find the value of X and prove the following​

Answer 1

Answer:

In △AXY and △ABC

∠XAY=∠BAC (Common angle)

∠AXY=∠ABC (Corresponding angles for parallel lines XY and BC)

∠AYX=∠ACB (Corresponding angles for parallel lines XY and BC)

Thus, △AXY∼△ABC (AAA rule)

Hence,

AB/AX = AY/AC = XY/BC

(Corresponding sides)

3+1.5/3 = XY/6

XY= 4.5/3×6

XY=4cm

Answer 2

The value of the variable x can be determined using the circle theorem to

determine the angle formed by x.

The correct responses are;

• (1) x = 30°

• (2) x = 100°

• (3) x = 60°

Reasons:

(1) From circle theorem, we have;

• Equal chords subtends equal arcs and vice versa

• Equal chords subtend equal angles at the center of a circle

Therefore;

Angle subtended by $\displaystyle \widehat {AB}$ at center = Angle subtended by $\widehat {BC}$ at the center;

• Angle at the center = 2 × Angle at the circumference

Therefore;

Angle subtended by $\displaystyle \widehat {AB}$ at the circumference = Angle subtended by $\mathbf{\widehat {BC}}$ at the circumference

Which gives;

• x = 30°

(2) $\displaystyle \widehat {AB} = \mathbf{\widehat {CD}}$ given

Angle subtended by arc $\widehat {CD}$ at the center = x

Equal arcs subtend equal angles at the center of a circle.

Therefore;

Angle subtended by arc $\mathbf{\widehat {AB}}$ at the center = x

Angle at center = 2 × angle at circumference

Therefore;

x = 2 × 50° = 100°

x = 100°

(3) $\displaystyle \widehat {AC} = \mathbf{ 3 \times \widehat {AB}}$ given

By definition, we have;

Angle subtended by $\widehat {AC}$ at the center = 3 × Angle subtended by $\widehat {AB}$ at the center

According to circle theorem, we have;

Angle at the center = 2 × Angle at the circumference

Let y represent the angle subtended at the circumference by $\widehat {AB}$, we have;

y = 20°

$\widehat {AB}$ = 2·y

$\widehat {AC}$ = 3 × $\widehat {AB}$ = 3 × 2·y = 6·y

Therefore;

$\widehat {AC}$ = 6 × 20° = 120°

$\widehat {AC}$ = 120°

$\widehat {AC}$ = 2·x

Therefore;

2·x = 120°

$\displaystyle x = \frac{120^{\circ}}{2} = \mathbf{ 60^{\circ}}$

x = 60°

Learn more circle theorems here:

brainly.com/question/16879446