Question

If a=(2+73)/(2-V3), b=(2-
73)/(2+13),then the value of a+b​

Answer 1

Answer:

Proof : /\ ABC and /\ DBC

seg AB seg DB

___________

angle ABC segment Angle DBC________

seg BC seg BC_______

.. A ABC = A DBC________

Fig. 3.30

... angle BAC angleBDC (c.a.c.t.)

In A ABC, ZBAC = 60° . ZBDC = 60°

ZDAC = ZADC = ZACD = 60°... sum of angles of A ADC is 180°

.: AADC is an equilateral triangle.

.. AC = AD = DC ........ corollary of converse of isosceles triangle theorem

1

But AB = AD........ construction

2

1

..AB AC ........ * AD= AC

2

1 / 2Theorem: If the acute angles of a right angled triangle have measures 30° and 60°, then

the length of the side opposite to 30° angle is half the length of the hypotenuse,

(Fill in the blanks and complete the proof.)

Given : In A ABC

ZB = 90°, ZC = 30°, ZA= 60°

60

To prove : AB

-

AC

30

B

C С

Fig. 3.29

Construction : Take a point D on the extended

seg AB such that AB = BD. Draw seg DC.

Proof : /\ ABC and /\ DBC

seg AB seg DB

___________

angle ABC segment Angle DBC________

seg BC seg BC_______

.. A ABC = A DBC________

Fig. 3.30

... angle BAC angleBDC (c.a.c.t.)

In A ABC, ZBAC = 60° . ZBDC = 60°

ZDAC = ZADC = ZACD = 60°... sum of angles of A ADC is 180°Kk

.: AADC is an equilateral triangle.

.. AC = AD = DC ........ corollary of converse of isosceles triangle theorem

1

But AB = AD........ construction

2

1

..AB AC ........ * AD= AC

2

1 / 2Proof : /\ ABC and /\ DBC

seg AB seg DB

___________

angle ABC segment Angle DBC________

seg BC seg BC_______

.. A ABC = A DBC________

Fig. 3.30

... angle BAC angleBDC (c.a.c.t.)

In A ABC, ZBAC = 60° . ZBDC = 60°

ZDAC = ZADC = ZACD = 60°... sum of angles of A ADC is 180°

.: AADC is an equilateral triangle.

.. AC = AD = DC ........ corollary of converse of isosceles triangle theorem

1

But AB = AD........ construction

2

1

..AB AC ........ * AD= AC

2

1 / 2Theorem: If the acute angles of a right angled triangle have measures 30° and 60°, then

the length of the side opposite to 30° angle is half the length of the hypotenuse,

(Fill in the blanks and complete the proof.)

Given : In A ABC

ZB = 90°, ZC = 30°, ZA= 60°

60

To prove : AB

-

AC

30

B

C С

Fig. 3.29

Construction : Take a point D on the extended

seg AB such that AB = BD. Draw seg DC.

Proof : /\ ABC and /\ DBC

seg AB seg DB

___________

angle ABC segment Angle DBC________

seg BC seg BC_______

.. A ABC = A DBC________

Fig. 3.30

... angle BAC angleBDC (c.a.c.t.)

In A ABC, ZBAC = 60° . ZBDC = 60°

ZDAC = ZADC = ZACD = 60°... sum of angles of A ADC is 180°Kk

.: AADC is an equilateral triangle.

.. AC = AD = DC ........ corollary of converse of isosceles triangle theorem

1

But AB = AD........ construction

2

1

..AB AC ........ * AD= AC

2

1 / 2Proof : /\ ABC and /\ DBC

seg AB seg DB

___________

angle ABC segment Angle DBC________

seg BC seg BC_______

.. A ABC = A DBC________

Fig. 3.30

... angle BAC angleBDC (c.a.c.t.)

In A ABC, ZBAC = 60° . ZBDC = 60°

ZDAC = ZADC = ZACD = 60°... sum of angles of A ADC is 180°

.: AADC is an equilateral triangle.

.. AC = AD = DC ........ corollary of converse of isosceles triangle theorem

1

But AB = AD........ construction

2

1

..AB AC ........ * AD= AC

2

1 / 2Theorem: If the acute angles of a right angled triangle have measures 30° and 60°, then

the length of the side opposite to 30° angle is half the length of the hypotenuse,

(Fill in the blanks and complete the proof.)

Given : In A ABC

ZB = 90°, ZC = 30°, ZA= 60°

60

To prove : AB

-

AC

30

B

C С

Fig. 3.29

Construction : Take a point D on the extended

seg AB such that AB = BD. Draw seg DC.

Proof : /\ ABC and /\ DBC

seg AB seg DB

___________

angle ABC segment Angle DBC________

seg BC seg BC_______

.. A ABC .

.. AC = AD = DC ........ corollary of converse of isosceles triangle theorem

1

But AB = AD........ construction

2

1

..AB AC ........ * AD= AC

2

1 / 2Theorem: If the acute angles of a right angled triangle have measures 30° and 60°, then

the length of the side opposite to 30° angle is half the length of the hypotenuse,

(Fill in the blanks and complete the proof.)

Given : In A ABC

ZB = 90°, ZC = 30°, ZA= 60°

60

To prove : AB

-

AC

30

B

C С

Fig. 3.29

Construction : Take a point D on the extended

seg AB such that AB = BD. Draw seg DC.

Proof : /\ ABC and /\ DBC

seg AB seg DB

___________

angle ABC segment Angle DBC________

seg BC seg BC_______

.. A ABC = A DBC________

Fig. 3.30

... angle BAC angleBDC (c.a.c.t.)

In A ABC, ZBAC = 60° . ZBDC = 60°

ZDAC = ZADC = ZACD = 60°... sum of angles of A ADC is 180°Kk

.: AADC is an equilateral triangle.

.. AC = AD = DC ........ corollary of converse of isosceles triangle theorem

1

But AB = AD........ construction

2

1

..AB AC ........ * AD= AC

2

1 / 2

Answer 2

$a = \frac{2 + 73}{2 - 13} = - \frac{75}{11} \\ \\ b = \frac{2 - 73}{2 + 13 } = - \frac{71}{15} \\ \\ \implies \: a + b = - \frac{75}{11} +( - \frac{71}{15} ) \\ \\ \implies \: a + b = - \frac{ 1125}{165} - \frac{781}{165} \\ \\ \implies a + b = - \frac{( 1125 - 781)}{165} \\ \\ \implies \underline{ \underline{ a + b = - \frac{344}{165} }}$