The perimeter of a quadrilateral is 56 cm56 cm. If the first three sides of a quadrilateral, taken in order are 17 cm,16 cm17 cm,16 cm, and 15 cm15 cm respectively, and the angle between fourth side and the third side is a right angle, find the area of the quadrilateral.
Answer:
Area: 180 cm2180 cm2
- The following picture shows the quadrilateral ABCDABCD,
The perimeter of the quadrilateral ABCD=56 cmABCD=56 cm
Therefore ,AB+BC+CD+DA=56⟹17+16+15+DA=56⟹48+DA=56⟹DA=56−48⟹DA=8 cm - Let's draw a line AC.
√DA2+DC2
The △ACD is the right angled triangle.
Therefore, AC2=DA2+DC2⟹AC=√DA2+DC2=√(8)2+(15)2=17 cm - The area of the right angled triangle △ACD=DA×DC2=8×152=60 cm2
- Now, we can see that, this quadrilateral consists of the triangles △ACD and △ABC.
The area of the △ABC can be calculated using Heron's formula since all sides of the triangle are known.
S=AB+BC+CA2=17+16+172=25 cm.
The area of the △ABC=√S(S−AB)(S−BC)(S−CA)=√25(25−17)(25−16)(25−17)=120 cm2 - The area of the quadrilateral ABCD=Area(△ACD)+Area(△ABC)=60+120=180 cm2