12. In the figure, ABC is a right-angled triangle. Find
(i) the area of ABC
(ii) the length of AC
(iii) the length of BD correct
Answers
Answer:
BD=8.78 cm
(1). The area of the triangle A = 180 cm^{2}cm
2
(2). The length of AC is = 41 cm
(3). The length of BD = 8.78 cm
Step-by-step explanation:
Given data
AB = 9 cm
BC = 40 cm
Since ABC is a right angle triangle.So
(1). Area of the triangle is = 0.5 × AB × BC
⇒ A = 0.5 × 9 × 40
⇒ A = 180 cm^{2}cm
2
Therefore the area of the triangle A = 180 cm^{2}cm
2
(2). From pythagoras theorm
\begin{gathered}AC^{2} = AB^{2} + BC^{2} \\\end{gathered}
AC
2
=AB
2
+BC
2
AC^{2} = 9^{2} + 40^{2}AC
2
=9
2
+40
2
AC^{2} = 1681AC
2
=1681
AC = 41 cm
Therefore the length of AC is = 41 cm
(3). Let AD = x & CD = 41 - x
From Δ ABD
BD^{2} = AB^{2} - AD^{2}BD
2
=AB
2
−AD
2
BD^{2} = 9^{2} - x^{2}BD
2
=9
2
−x
2
BD^{2} = 81 - x^{2}BD
2
=81−x
2
------- (1)
From Δ BDC
BD^{2} = BC^{2} - CD^{2}BD
2
=BC
2
−CD
2
BD^{2} = 40^{2} - (41 - x)^{2}BD
2
=40
2
−(41−x)
2
BD^{2} = -81 - x^{2} + 82 xBD
2
=−81−x
2
+82x ------- (2)
Equation 1 = Equation 2
⇒ 81 - x^{2} = - 81 - x^{2} + 82 x81−x
2
=−81−x
2
+82x
⇒ 82 x = 16282x=162
⇒ x = 1.976 cm
We know that x = AD = 1.976 cm
Put the value of x in equation 1 we get
BD^{2} = 81 - 1.976^{2}BD
2
=81−1.976
2
BD^{2}BD
2
= 77.095
BD = 8.78 cm
Therefore the length of BD = 8.78 cm