A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that BD = 8 cm and DC = 6 cm. Find the sides AB and AC.
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Step-by-Step Solution
Step 1: Identify Tangent Properties
According to the property of tangents drawn from an external point to a circle, the lengths of the tangents from the external point to the circle are equal. Here, the vertices A, B, and C are external points. So, BD=BF, CD=CE, and AE=AF. We are given BD=8 cm and CD=6 cm. Let $AE = AF = x.
Step 2: Express Sides in terms of x
Now we can express the lengths of the sides of the triangle ABC in terms of x. Side AB is the sum of AF and FB, which is x+8. Side AC is the sum of AE and EC, which is x+6. Side BC is the sum of BD and DC, which is 8+6=14 cm.
Step 3: Calculate Semi-perimeter (s)
The semi-perimeter s of the triangle is half the sum of its sides. Substituting the expressions for AB, BC, and AC, we get s=2(x+8)+14+(x+6), which simplifies to s=x+14.
Step 4: Calculate Area using Heron's Formula
We use Heron's formula to calculate the area of triangle ABC. Substituting the values of s, AB, BC, and AC into the formula s(s−AB)(s−BC)(s−AC), we simplify the expression to 48x(x+14).
Step 5: Calculate Area using Radius and Semi-perimeter
Alternatively, the area of a triangle that circumscribes a circle can also be calculated using the formula Area=r⋅s, where r is the inradius and s is the semi-perimeter. Given the radius r=4 cm and s=x+14, the area is $4(x+14).
Step 6: Equate Areas and Solve for x
By equating the two expressions for the area of triangle ABC, we get 48x(x+14)=4(x+14). Squaring both sides and simplifying, we arrive at 3x=x+14, which gives 2x=14, so x=7. We can divide by (x+14) because x represents a length, so x>0, making x+14 =0.
Step 7: Calculate Sides AB and AC
Finally, substitute the value of x=7 back into the expressions for AB and AC. This gives AB=7+8=15 cm and AC=7+6=13 cm.