20) The diagonals of a rhombus measure 16 cm and 12 cm. Find the perimeter of the
rhombus.
21) In an isosceles triangle ABC, angle ABC = 45° and BC = CA, then calculate angle C.
Answers
Answer:
20. The perimeter of the rhombus is 40 cm.
21. The measure of angle C of the isosceles triangle is 90°.
Step-by-step-explanation:
NOTE: Refer to the attachments for the diagrams.
20.
We have given that,
The diagonals of a rhombus are 16 cm & 12 cm.
We have to find the perimeter of the rhombus.
In figure, □ABCD is a rhombus.
Seg AC & Seg BD are the diagonals of the rhombus.
The diagonals AC & BD intersect at point O.
AC = 16 cm
BD = 12 cm - - - [ Given ]
Now, we know that,
Diagonals of a rhombus bisect each other.
∴ AO = OC = ½ * AC
⇒ AO = OC = ½ * 16
⇒ AO = OC = 8 cm - - ( 1 )
Now,
BO = OD = ½ * BD
⇒ BO = OD = ½ * 12
⇒ BO = OD = 6 cm - - ( 2 )
Now, we know that,
Diagonals of a rhombus are perpendicular bisectors of each other.
∴ AC ⊥ BD
Now, in △AOD, m∟AOD = 90°
∴ ( AD )² = ( AO )² + ( OD )² - - [ Pythagoras theorem ]
⇒ ( AD )² = ( 8 )² + ( 6 )² - - [ From ( 1 ) & ( 2 ) ]
⇒ ( AD )² = 64 + 36
⇒ ( AD )² = 100
⇒ AD = √100 - - [ Taking square roots ]
∴ AD = 10 cm
∴ Side of the rhombus is 10 cm.
Now, we know that,
Perimeter of rhombus = 4 * side
⇒ P ( □ABCD ) = 4 * AD
⇒ P ( □ABCD ) = 4 * 10
⇒ P ( □ABCD ) = 40 cm
∴ The perimeter of the rhombus is 40 cm.
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21.
In figure, △ABC is an isosceles triangle.
m∠ABC = 45°
BC = CA - - [ Given ]
Now, we know that,
Angles opposite to congruent sides are also congruent.
∴ m∠ABC = m∠BAC
⇒ 45° = m∠BAC
∴ m∠BAC = 45°
Now, we know that,
The sum of measures of angles of a triangle is 180°.
∴ In △ABC,
m∠ABC + m∠BAC + m∠ACB = 180°
⇒ 45° + 45° + m∠ACB = 180°
⇒ 90° + m∠ACB = 180°
⇒ m∠ACB = 180° - 90°
⇒ m∠ACB = 90°
∴ The measure of angle C of the isosceles triangle is 90°.
