Answer:
47.62 cm
Step-by-step explanation:
ABCD is a rhombus. Each side measures 15 cm.
m<A = 60°.
Here are some properties of a rhombus:
Opposite angles of a rhombus are congruent.
m<C = m<A = 60°
m<ADC = m<ABC
The sum of the measures of the interior angles of a quadrilateral is 360°.
The diagonals are perpendicular bisectors of each other.
m<A + m<C + m<ADC + m<ABC = 360°
60° + 60° + 2m<ADC = 360°
2m<ADC = 240°
m<ADC = 120°
Each diagonal of a rhombus bisects a pair of congruent, opposite angles.
m<ADB + m<CDB = m<ADC
m<ADB = m<CDB
m<ADB + m<ADB = 120°
m<ADB = 60°
Triangle ABD had two angles, <A and <ADB, that measure 60°. Therefore, the third angle, <ABD also measures 60°. That makes triangle ABD an equilateral triangle.
Draw segment AT.
T is the midpoint of BD. The segment AT is the perpendicular bisector of segment BD. Triangle ATD is a 30-60-90 triangle.
Using the ratio of the lengths of the sides of a 30-60-90 triangle, we can calculate the length of AT which is the radius of circle A.
TD : AT : AD
1 : √3 : 2
AD = 15 cm
TD = AD/2
AT = √3 × TD
AT = (AD√3)/2
AT = 7.5√3 = 12.99
AQ = 12.99
perimeter = QD + PB + m(arc)PTQ + BC + CD
perimeter = (15 - 12.99) + (15 - 12.99) + 60°/360° × 2π(12.99) + 15 + 15
perimeter = 47.62
Answer: perimeter = 47.62 cm
