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Two right-angled triangles are drawn inside a circle such that their bases are on the same diameter of the circle. Their hypotenuse is combined to form a chord of the circle. Find the total area of two triangles if the radius of the circle is 2.
Step 1:
Join ends of the chord (Vertices of triangles) to the center of the circle to form radiuses of that circle.
Srep 2:
Let AB = BC = a;
and CD = DE = b;
also OC = x
⇒ BO = a - x;
and OD = b + x;
AO = EO = Radius = 2;
Step 3:
Observe that triangles ABO and ODE are right-angle triangles.
⇒ AB² + BO² = AO²
⇒ (a)² + (a - x)² = 2²
⇒ a² + a² - 2ax + x² = 4
⇒ 2a² - 2ax + x² = 4 -------(1)
Also,
⇒ OD² + DE² = OE²
⇒ (b+x)² + (b)² = 2²
⇒ b² + 2bx + x² + b² = 4
⇒ 2b² + 2bx + x² = 4 -------(2)
From (1) and (2):
2a² - 2ax + x² = 2b² + 2bx + x²
⇒ a² - ax = b² + bx
⇒ a² - b² = ax + bx
⇒ (a + b)(a - b) = (a + b)x
⇒ x = a - b
Step 4:
Now BO = a - x = b
AB² + BO² = AO²
⇒ a² + b² = 4
Sum of areas of two triangles = (a² + b²)/2 = 2
Image Solution:
Solution 1
Solution 2
Question related to Geometry, Circle, Triangle, Area.
